RELATIONSHIPS BETWEEN LANE CHANGE PERFORMANCE and OPEN- LOOP HANDLING METRICS Robert Powell Clemson University, [email protected]

RELATIONSHIPS BETWEEN LANE CHANGE PERFORMANCE and OPEN- LOOP HANDLING METRICS Robert Powell Clemson University, Rpowell@G.Clemson.Edu

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1-1-2009 RELATIONSHIPS BETWEEN LANE CHANGE PERFORMANCE AND OPEN- LOOP HANDLING METRICS Robert Powell Clemson University, [email protected]

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Recommended Citation Powell, Robert, "RELATIONSHIPS BETWEEN LANE CHANGE PERFORMANCE AND OPEN-LOOP HANDLING METRICS" (2009). All Theses. Paper 743.

This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. RELATIONSHIPS BETWEEN LANE CHANGE PERFORMANCE AND OPEN-LOOP HANDLING METRICS

A Thesis Presented to the Graduate School of Clemson University

In Partial Fulﬁllment of the Requirements for the Degree Master of Science Mechanical Engineering

by Robert A. Powell December 2009

Accepted by: Dr. E. Harry Law, Committee Co-Chair Dr. Beshahwired Ayalew, Committee Co-Chair Dr. John Ziegert Abstract

This work deals with the question of relating open-loop handling metrics to driver- in-the-loop performance (closed-loop). The goal is to allow manufacturers to reduce cost and time associated with vehicle handling development. A vehicle model was built in the

CarSim environment using kinematics and compliance, geometrical, and ﬂat track tire data.

This model was then compared and validated to testing done at Michelin’s Laurens Proving

Grounds using open-loop handling metrics. The open-loop tests conducted for model validation were an understeer test and swept sine or random steer test. Four commonly used handling metrics (steady state yaw rate gain, yaw rate damping ratio, yaw rate bandwidth, and lateral acceleration phase lag at 1 Hz) were extracted from the frequency response functions of the swept sine test. These are the open-loop handling parameters used to draw relationships to closed-loop performance.

Next, a driver model was coupled to the vehicle model in order to simulate a closed- loop maneuver. Quadratic cost functions are then introduced as a means to measure performance through the closed-loop ISO Double Lane Change maneuver. These quadratic metrics measure path-following ability, and the mental and physical workload of the driver.

Driver model parameters were determined by weighting the quadratic cost functions to select the optimum driver with the lowest total cost. In this work, highest priority was given to path-following ability in order to successfully complete the lane change without violating course boundaries. Mental workload and physical workload were given lower priority because of the short length of the maneuver.

As a means to change the vehicle model, three hypothetical tires are introduced. To

ii have an even greater number of configurations, the vehicle model’s static weight distribution was altered. This yielded twelve different realistic vehicle configurations from which to draw conclusions. With the twelve different vehicle models considered, qualitative relationships were found between open-loop handling measures and driver-in-the-loop quadratic performance metrics. Two relationships were found to exist–when steady state yaw rate gain and yaw rate bandwidth are increased, path-following ability is enhanced. When yaw rate bandwidth is increased and lateral acceleration phase lag at 1 Hz is decreased, mental and physical workload are reduced on the driver.

iii Acknowledgments

I would ﬁrst like to thank my advisors, Dr. E. Harry Law and Dr. Beshahwired

Ayalew, for their wisdom and guidance in helping me achieve this goal. I would also like to thank Dr. John Ziegert for being a part of my committee and providing academic assistance in my graduate course work. In addition, I would like to thank Lynn Vorys for her help in the editing process of this document.

A special thanks also goes to Michelin America’s Research Center and BMW for providing the funding for this project. In addition, I would like to thank Dr. Tim Rhyne and Dr. Steve Cron of Michelin and Dr. Andreas Obieglo of BMW for their continued support throughout this research.

iv Dedication

I would like to dedicate this thesis to my parents, David and Leslie, who have always believed in me and helped me achieve my goals. In addition, I would like to thank the rest of my family for their unwavering support throughout my educational career.

To my friends and oﬃce mates, thank you for making my experience at Clemson both enjoyable and educational. A special thanks goes to Anup Khekare with whom I have had the pleasure of working on this research.

v Table of Contents

Abstract ...... ii

Acknowledgments ...... iv

Dedication ...... v

List of Figures ...... ix

List of Tables ...... xi

1 Introduction ...... 1 1.1 Introduction ...... 1

1.2 Research Motivation and Problem Statement ...... 2

1.3 Literature Review ...... 4

1.3.1 Open-Loop Handling Metrics ...... 4

1.3.2 Relation Between Subjective and Objective Testing ...... 4

1.3.3 Relationship Between Open and Closed-Loop Testing ...... 5

1.3.4 Driver Models ...... 6

2 Vehicle Models and Open-Loop Tests ...... 8 2.1 Introduction ...... 8

2.2 Vehicle Model ...... 9

2.3 Open-Loop Handling Tests ...... 10

2.3.1 Swept Sine Test ...... 11

vi 2.3.2 Understeer Test ...... 12

2.4 Validation of CarSim Model ...... 13

3 Closed-Loop Modeling and Metrics ...... 16 3.1 Introduction ...... 16

3.2 Driver Model A ...... 16

3.3 ISO Double Lane Change Course and Performance Metrics ...... 18

3.3.1 ISO Double Lane Change Course ...... 18

3.3.2 Quadratic Cost Functions ...... 20

3.4 Driver Parameter Selection ...... 22

3.5 Selection of Tire Characteristics ...... 29

4 Results and Discussion ...... 32 4.1 Introduction ...... 32

4.2 Open-Loop and Closed-Loop Performance with Changing Tires ...... 32

4.3 Open-Loop and Closed-Loop Performance Metrics with Changing Weight

Distribution ...... 40

4.4 Qualitative Analysis Relating Open and Closed-Loop Handling Metrics . . 43

5 Conclusions and Future Work ...... 48 5.1 Conclusions ...... 48

5.2 Future Work ...... 49

A Vehicle Model Parameters ...... 51 A.1 Vehicle Properties ...... 52

A.2 Suspension Parameters ...... 53

A.3 Suspension Kinematic Parameters ...... 54

A.4 Compliance Parameters ...... 58

A.5 Tire Parameters ...... 62

A.6 Kingpin Geometry ...... 63

vii B Driver Models ...... 64 B.1 Driver Model A ...... 64

B.2 Driver Model B ...... 67

B.3 Driver Model C ...... 70

C Methodology for Creating the TWEELs™ ...... 74

D Open and Closed-Loop Raw Data ...... 78

E CarSim File Formats ...... 81

F MATLAB Programs ...... 84 F.1 Model_Validation.m ...... 84

F.2 TWEEL_Lateral_Force.m ...... 93

F.3 TWEEL_Aligning_Moment.m ...... 105

F.4 Spider Plot Programs ...... 112

F.4.1 color_index.m ...... 112

F.4.2 isint.m ...... 113

F.4.3 spider.m ...... 114

F.5 Frequency_Response_Functions.m ...... 122

F.6 Lane_Change.m ...... 155

References ...... 167

viii List of Figures

1.1 Image of Generic TWEEL™ ...... 3

2.1 Transfer Function for RWA/SWA ...... 10

2.2 Frequency Response Functions (LPG Test vs. CarSim Simulation) . . . . . 14

2.3 Spider Chart Comparing CarSim Simulation vs. LPG Test ...... 15

2.4 Understeer Test Comparing CarSim Simulation vs. LPG Test ...... 15

3.1 Genta driver and vehicle closed loop system ...... 18

3.2 ISO Lane Change Course ...... 20

3.3 Yerror vs. Time for each J’s Lowest Cost ...... 24 3.4 SWA vs. Time for each J’s Lowest Cost ...... 26

3.5 SWA Rate vs. Time for each J’s Lowest Cost ...... 27

3.6 Jtotal for OE Tire at 78 kph ...... 28 3.7 Cornering Stiﬀness versus Vertical Load for TWEELs™ ...... 30

3.8 Lateral Force vs. Slip Angle for OE Tire ...... 31

4.1 Frequency Response Functions Comparing OE Tire to TWEELs™ . . . . . 33

4.2 Spider Plot of Open-Loop Handling Metrics with Diﬀerent Tires ...... 34

4.3 Vehicle Trajectory with OE Tire at Maximum Speed of 78 kph ...... 36

4.4 Vehicle Response Parameters vs. Time at 78 kph ...... 37

4.5 Lateral Acceleration vs. Time and Distance at 78 kph ...... 38

4.6 Lateral Position and Heading Angle vs. Time at 78 kph ...... 39

4.7 Frequency Response Functions Comparing Weight Distributions ...... 41

ix 4.8 Spider Plot of Open-Loop Handling Metrics for Changing Weight Distribution 42

4.9 Relating Select Open and Closed-Loop Metrics in 2-D Plots ...... 44

4.10 3-D Plot Comparing SS Yaw Rate Gain and Yaw Rate Bandwidth to J1 . . 45

4.11 3-D Plot Comparing Lateral Accel. Phase Lag and Yaw Rate Bandwidth to J2 46

4.12 3-D Plot Comparing Lateral Accel. Phase Lag and Yaw Rate Bandwidth to J3 47

B.1 Original Genta Driver Model Block Diagram ...... 64

B.2 Driver Model A Simulink/CarSim Block Diagram ...... 66

B.3 Driver Model B Simulink/CarSim Block Diagram ...... 69

B.5 Driver Model C Simulink/CarSim Block Diagram ...... 72

B.4 Horiuchi Driver/Vehicle Closed Loop System ...... 73

C.1 BMW 3 Series Aligning Moment Used for Mini Cooper in CarSim . . . . . 77

E.1 Sample ERD ﬁle ...... 82

x List of Tables

3.1 J1 at 78 kph Normalized to Lowest Value ...... 24

3.2 J2 at 78 kph Normalized to Lowest Value ...... 25

3.3 J3 at 78 kph Normalized to Lowest Value ...... 25

3.4 Jtotal for OE Tire at 78 kph ...... 28 3.5 Driver Parameters for Driver Model A ...... 28

3.6 Cornering Stiﬀness for each TWEEL™ ...... 29

4.1 Spider Chart Data for TWEELs™ ...... 35

4.2 Quadratic Cost Functions at Each Conﬁguration’s Maximum Speed . . . . 35

4.3 Spider Chart Data for Changing Weight Distribution ...... 40

4.4 Quadratic Cost Functions at Each Conﬁguration’s Maximum Speed . . . . 42

4.5 Legend for Figures in Current Section ...... 43

A.1 Vehicle Properties ...... 52

A.2 Suspension Parameters ...... 53

A.3 Left Front Suspension Kinematic Parameters ...... 54

A.4 Right Front Suspension Kinematic Parameters ...... 55

A.5 Left Rear Suspension Kinematic Parameters ...... 56

A.6 Right Rear Suspension Kinematic Parameters ...... 57

A.7 Left Front Compliance Parameters ...... 58

A.8 Right Front Compliance Parameters ...... 59

A.9 Left Rear Compliance Parameters ...... 60

A.10 Right Rear Compliance Parameters ...... 61

xi A.11 Tire Parameters ...... 62

A.12 Kingpin Geometry Parameters ...... 63

D.1 4 Parameter Data for 12 Conﬁgurations ...... 79

D.2 Raw Values of Quadratic Cost Functions at 77 kph ...... 79

D.3 Normalized Values of Quadratic Cost Functions at 77 kph ...... 80

xii Chapter 1

Introduction

1.1 Introduction

Open-loop testing is an established practice which engineers use to objectively measure handling qualities of a vehicle. By deﬁnition, the driver should not provide any correc- tive action in open-loop testing thus making the tests repeatable and not dependent upon the driver. Although standards exist for how to conduct the test, there are no standards deﬁning acceptable values for the extracted handling metrics [7]. These tests are used to establish relationships between changes being made to the vehicle and its handling properties. Mimuro et al. [2] suggest arranging commonly used handling metrics in a spider chart to depict trends in behavior. The four parameters Mimuro et al. suggested are steady state yaw rate gain, yaw rate natural frequency, yaw rate damping ratio, and lateral acceleration phase lag at 1 Hz.

While general ideas exist on which parameters improve open-loop performance, the link between open-loop handling metrics and driver-in-the-loop (i.e., closed-loop) performance is still relatively unknown. When the driver is in the loop it is more difficult to relate driver evaluation to specific and quantifiable handling metrics and thus to vehicle design. Crolla et al. point out that car manufacturers employ highly skilled drivers to subjectively measure vehicle handling, but this process is both inefficient and costly [7].

1 The remaining part of this chapter discusses the motivation behind researching this topic and reviews the previous work done in this field. Chapter 2 presents the vehicle model used and the open-loop tests conducted to validate the vehicle model. The driver model that was adopted and the closed-loop maneuver is explained along with the metrics used to measure performance in the maneuver in Chapter 3. In the same chapter, the different tires fitted to the vehicle are explained. Chapter 4 presents the results for both open-loop and closed-loop testing and the conclusions that were drawn regarding the relationships between the two. Finally, results will be summarized and possible extensions of this work suggested. Additional support information is included in the appendices.

1.2 Research Motivation and Problem Statement

Prior to marketing any new car, manufacturers must spend large amounts of time and money developing vehicles. This requires building prototype vehicles and carrying out many tests. Highly-skilled professional test drivers are used to evaluate vehicle handling and ride comfort. If the professional driver does not sign-oﬀ on the vehicle then the designer must address the driver’s concerns before the vehicle can proceed in its development cycle.

This method of vehicle development is both costly and ineﬃcient.

This thesis addresses the method by which vehicle handling is evaluated. In this thesis, it is desired to relate open-loop handling metrics, which are easily modeled and simulated, to driver-in-the-loop performance. Relating these two types of testing methods will allow an engineer to produce a design which will need less iterations to receive a sign-off by the test driver thereby reducing costs and development time. In order to infer relationships, many different vehicle configurations are considered.

In this work, different vehicle configurations are achieved by altering the tires fitted to a single model. In this thesis, the new tires fitted to the vehicle are hypothetical and meant to replicate the handling characteristics of the TWEEL™ (a non-pneumatic tire) developed by Michelin. A generic image of a TWEEL™ is shown in Figure 1.1.

2 Figure 1.1: Image of Generic TWEEL™

3 1.3 Literature Review

As stated by Crolla et al., [7] there are deﬁned standards for conducting open-loop and subjective testing. Subjective testing is practiced quite extensively in the automotive industry and manufacturers design towards established general ranges of performance. The major problem is that there is no link between closed-loop and open-loop testing. If there were, manufacturers could realize signiﬁcant time and money savings.

1.3.1 Open-Loop Handling Metrics

One of the more commonly cited papers for open-loop testing is by Mimuro et al. [2] in which they suggest using a pulse steer at 100 kilometers per hour (kph) to develop transfer functions for yaw rate and lateral acceleration versus steering wheel angle.

From these transfer functions, four parameters are extracted which Mimuro et al. contend correlate well with subjective driver ratings. The four parameters are lateral acceleration phase lag at 1 Hz, steady state yaw rate gain, yaw rate natural frequency, and yaw rate damping ratio. To present these four parameters in a logical and easy-to-read way, Mimuro et al. arrange them in a “rhombus” or spider plot. Presenting it in this way allows the reader to easily compare the handling characteristics of multiple vehicles simultaneously.

1.3.2 Relation Between Subjective and Objective Testing

In order to reduce cost and development time, research has been done to relate subjective and objective handling metrics. In a paper by Crolla et al. [7], many diﬀerent open-loop handling metrics are measured and correlated to subjective driver ratings. There were sixteen vehicle conﬁgurations with forty-nine objective handling metrics measured.

These open-loop tests included a steady state understeer test, step steer input (J-turn), and impulse (frequency response) test. For the steady state understeer test, some of the parameters measured were lateral acceleration versus steering wheel angle, side slip angle versus lateral acceleration, and roll angle. In the J-turn test the parameters measured in-

4 cluded peak lateral acceleration response time, peak road wheel steer angle and response time, and peak yaw rate and response time. The authors then asked eight experienced test drivers forty-nine questions about each of the sixteen vehicle conﬁgurations. The answers were then averaged among the eight test drivers and a linear regression was applied. Accord- ing to the authors, while this provides some good insight into the links between subjective and objective ratings, there is still no clear link between the two. This paper is clearly a step in the right direction for ﬁnding a link between objective and subjective testing.

In a later paper by King et al.[8], a neural network is used to correlate subjective and objective handling ratings. This paper builds oﬀ of the earlier paper by Crolla et al.

[7] by using the same test data for both objective testing and driver assessments. The main diﬀerence in the two is the method of correlation. In this paper by King, a neural network is used to anticipate which parameters drivers will like better. The neural networks provided non-linear links between subjective and objective handling metrics. The problem with this method that the authors point out is that, in order to function properly, the neural network uses some of the gathered data for learning. This results in fewer data points to predict subjective ratings given the objective inputs. However, it is encouraging that the ranges provided by the authors for many open-loop parameters correlated well with subjective ratings .

1.3.3 Relationship Between Open and Closed-Loop Testing

Bergman [11] uses several different tests to try to draw a link between open-loop and closed-loop tests. For the study, two different Ford production sedans were used in three different configurations: curb plus driver (C+D), gross vehicle weight (GVW), and GVW with reduced rear tire inflation pressure. The three open-loop maneuvers employed were step steer (J-turn), modified step steer, and reverse step steer. The step steer requires a driver to travel in a straight line and then input a predefined steering wheel angle as quickly as possible. The modified step steer consists of the vehicle traveling along a slightly curved path then the driver inputs the step steer in the same direction the vehicle is traveling.

5 The reverse step steer consisted of the vehicle following a slightly curved path in a given direction and then suddenly giving a step steer input in the opposite direction of the vehicle path. Three different lateral accelerations of 0.15g, 0.30g and 0.45g were achieved for each test. The closed-loop maneuver was a single lane change at the same levels of lateral acceleration as the open-loop tests. Bergman finds a good linear correlation between open- loop lateral acceleration response time and closed-loop lateral acceleration response time using the reverse step steer. He further states that the results are strongly influenced by the particular response parameter selected and the test procedure considered.

In a survey, Good reviewed the parameters that correlate well among open-loop, closed-loop, and subjective testing. The papers examined various subjects including ac- cident avoidance, driver-vehicle performance, driver strategy for negotiating a maneuver, open-loop testing, closed-loop testing, and subjective testing.

In another study that Good reviewed [12], he noted that the authors, McRuer and

Klien [13], cite the eﬀects of steering ratio on closed-loop performance with an expert driver and three “representative” drivers. The interesting observation from this study is that the expert driver changed his gain as the steering ratio of the vehicle changed while the

“representative” drivers did not show the same behavior. Good also cites a paper by Lincke et al. [14] in which eight unskilled drivers negotiated a double lane change at 100 kph.

Lincke et al. found that the drivers’ subjective ratings correlated well with the natural frequency of the vehicle with damping ratio being much less important.

1.3.4 Driver Models

Another area of review in this thesis is driver models. There are many diﬀerent driver models put forward and each has its own beneﬁts and drawbacks. In this review, three will be examined. Two are proposed by Genta [15] (one a simple version of the other) and the third is proposed by Horiuchi and Yuhara [16].

In describing a driver model, Genta [15] describes some necessary aspects required in order to perform a constant speed task. These parameters are lead, lag, time delay, and

6 gain. He then gives a ﬁrst-order transfer function with each of these parameters (which will be explained in further detail in Chapter 3).

Proposing an even simpler driver model, Genta states that the lead term can be neglected and the time delay and lag can be combined as a single term. This yields a much simpler ﬁrst-order transfer function that is easily implemented in MATLAB/Simulink. He also adds to the model a term where the driver is previewing the error between the current path and the desired path ahead.

In the paper by Horiuchi and Yuhara [16], they propose a driver model that has three essential characteristics: time delay and time lag which are due to mental processing time and neuromuscular system lag, respectively; lead time or predictive action of the driver; and driver gain. The model proposed has a driver that uses lateral position command as an input to the system and then uses the lateral position error and the yaw angle error as feedback into the driver model. The driver outputs a steering wheel angle to negotiate the course. This paper further goes on to give metrics for determining driver workload and task performance. These will be discussed in a later chapter.

7 Chapter 2

Vehicle Models and Open-Loop Tests

2.1 Introduction

Data for the vehicle and tires are necessary to create a valid handling model. A full non-linear CarSim model is presented in this thesis which was used to capture higher order eﬀects such as the non-linearity of the tire characteristics and suspension kinematics.

Kinematics and compliance (K&C) measurements were performed on the vehicle (2007

BMW Mini Cooper S) at the Michelin Americas Research Center (MARC) to characterize all of the geometric and suspension characteristics. Michelin also performed ﬂat track testing on the original equipment (OE) tires and provided the results in the form of coeﬃcients which could be used in Pacejka’s Magic Formula [3]. Open-loop handling tests, including a swept sine and an understeer test of the vehicle, were completed at Michelin’s Laurens

Proving Grounds (LPG). In simulation, the same tests were performed on the CarSim model and compared to the results obtained at LPG in order to validate the CarSim vehicle model.

8 2.2 Vehicle Model

In order to have a high-ﬁdelity and comprehensive model, commercial software was used instead of developing an in-house model. CarSim, by Mechanical Simulation, is widely used by many companies including Michelin and requires tire and suspension data to be entered as look-up tables. The CarSim model is proprietary software that uses a system of approximately 55 DOF to model the vehicle [19] although some of these degrees of freedom are for things such as the engine and transmission which have no bearing on this analyses.

For the purposes of this thesis, only parameters aﬀecting lateral dynamics were used for the analyses. All of the suspension and tire data used for the CarSim model was measured at MARC and provided via the K&C report [4] and coeﬃcients used in Pacejka’s Magic

Formula [3, 5] for the tire lateral force and aligning moment curves as functions of vertical load, slip angle, and camber. Vehicle weights and CG locations for the test conﬁgurations were measured at LPG [20]. A detailed list of the required parameters and their source for the CarSim model is given in Appendix A.

CarSim also has a built-in steering system model requiring parameters such as steering inertia, hysteresis, and damping. However these parameters were not measured. As these steering parameters were not measured, testing at LPG was done not only by measuring steering wheel angle but also rack displacement. As it is more common to show handling parameters with respect to steering wheel angle (SWA), a method was devised to show CarSim output, initially calculated in road wheel angle, with respect to SWA. As track test data was recorded for both rack displacement and steering wheel angle, a transfer function was created between the two over the frequency range of interest as shown in

Figure 2.1. To get the yaw rate and lateral acceleration transfer functions in terms of SWA, the magnitudes are multiplied and the phases are added. An interesting note about Figure

2.1 is that the inverse steering ratio (RWA to SWA) is not constant over the frequency range. For an understeer test, which is considered a steady state test, a steering ratio of

17.4:1 SWA to RWA was used since this is the value at the steady state frequency for Figure

9 Figure 2.1: Transfer Function for RWA/SWA

2.1. Transfer functions were computed in CarSim by using the Spectrum Analyzer tool to perform a fast Fourier transform (FFT) on time history data from LPG and the simulation.

2.3 Open-Loop Handling Tests

Open-loop testing is widely used to objectively gather information about vehicle handling with the driver removed from the loop. Although a driver is still used to perform the test, he does so without regard to feedback from the vehicle. The two open-loop tests used at LPG and in simulation are a swept sine and an understeer test.

10 2.3.1 Swept Sine Test

A swept sine wave as a steering input gives frequency response information about the vehicle. The goal is to have as much frequency content as possible; however, in physical testing, on the high end of the frequency spectrum, a human driver is only capable of about

3 to 3.5 Hz of frequency input into the steering wheel. On the low end of the frequency spectrum, the test is usually limited by the length of the testing track because it takes a long time and, therefore, a long distance at the track to get the required low frequency content. Conversely, in simulation the maximum achievable frequency can be as high as needed but it is usually restricted to approximately 7-10 Hz. As far as the low frequency is concerned there are no limits to the length of the test track in simulation and it can be run as long as needed.

The measured parameters in this test are lateral acceleration, yaw rate, steering wheel angle and/or road wheel angle, and time. A fast Fourier transform (FFT) is computed with the time history data to transform it into frequency domain. Two transfer functions are computed from the time history data– yaw rate and lateral acceleration (Figure 2.2).

From these two transfer functions, the four parameters described by Mimuro et al. [2] can be extracted. When Mimuro et al, calculate the four parameters, they fit a two degree of freedom (2 DOF) system to test data and extract the four parameters. In this work, instead of fitting a 2 DOF system to the data, the four parameters are extracted directly from the transfer functions. This necessitates a slightly different definition of yaw damping ratio and use of yaw rate bandwidth as opposed to yaw rate natural frequency. Yaw rate damping ratio in this thesis is derived from the definition of magnitude ratio which is presented in

Equation 2.1.

1 1 |G| = s |ω=ωn = (2.1) 2 2 2ζ 1 − ω + 2ζ ω ωn ωn where G is the peak yaw rate gain divided by the steady state yaw rate gain, ωn is the

11 natural frequency, and ζ is the damping ratio.

Mimuro et al. suggest plotting these four parameters in a spider chart which shows handling tendencies. An example of the spider chart can be seen in Figure 2.3 which compares the LPG test results to the CarSim model.

In general, when the area of the rhombus for one vehicle is larger than another

(assuming consistent axes), the vehicle with the larger area is considered to have greater handling potential. It is important to note for this spider chart that when a rhombus shifts to the upper right-hand corner in relation to other rhombi, the vehicle tends towards more understeer. Whereas when a rhombus shifts toward the lower left, the vehicle tends towards less understeer. [2].

For the test at LPG on the Mini, Michelin provided both an objective handling report

[20] (in which they used in-house software to process the yaw rate and lateral acceleration transfer functions) and understeer test results. Michelin also provided the raw data which included ﬁve runs of each test to Clemson for processing. All runs were averaged (by

Michelin and in this thesis) in order to smooth out any irregularities in one particular test.

With the raw testing data, transfer functions could then be computed in CarSim’s Spectrum

Analyzer tool. This allowed for consistent processing of the data and results could be stated with conﬁdence.

2.3.2 Understeer Test

Another important test for determining handling characteristics is an understeer test. In simulation, this is done by having a vehicle start from a standstill and accelerate at a constant rate while traversing a 500 foot radius circle with a track coefficient of friction of 1.0. The actual tests conducted at LPG are slightly different than what is done in simulation. This is due to the fact that, in order to get reliable results and smooth out any irregularities with one test, multiple tests have to be run and averaged (in this case five runs). If a vehicle were to be run multiple times around a circular track at the speed required for this test it would cause excessive wear on the outside tires and therefore drastically alter

12 the handling properties of the vehicle. In simulation tire wear is not a concern therefore the test is simple to complete and process the results.

The most common data to observe from this test is steering wheel angle (SWA) versus lateral acceleration. Calculating the initial slope of the graph (approximately from 0.1g to 0.3g) will provide the understeer gradient. The understeer gradient gives an indication of how much lateral acceleration will be achieved for a given steering wheel angle. Figure 2.4 shows a comparison of LPG test data to that of the CarSim simulation for steering wheel angle versus lateral acceleration.

2.4 Validation of CarSim Model

Once the model was developed in CarSim, simulations were run similar to the tests conducted at LPG– specifically a swept sine test and an understeer test as described in the previous two sections. For the former test, transfer functions for yaw rate and lateral acceleration were calculated and compared to LPG test results. For comparing CarSim simulations to testing performed at LPG, all results are presented with respect to RWA in order to remove the steering system completely from validation of the model. The analyses in this document only examine the curb plus driver configuration as tested at LPG. The parameters used in the CarSim model are listed in Appendix A. Figure 2.2 shows the yaw rate and lateral acceleration transfer functions with respect to RWA. It can be seen from the figure that the transfer functions match well in the lower frequency range (less than 1.5

Hz). Low frequency results are expected to match well because kinematics and compliance data are taken in a quasi-static state. When the actual vehicle is excited at high frequencies it is less likely the simulation and LPG test results will match. Figure 2.3 is a spider chart extracting the four parameters explained by Mimuro et al. [2] comparing the CarSim simulation and LPG test results. From Figure 2.3 it is clear that two of the parameters

(steady state yaw rate gain and yaw rate damping ratio) are a perfect match to the LPG test data whereas the other two (yaw rate bandwidth and lateral acceleration phase lag at

13 1 Hz) remain close.

Figure 2.4 shows a comparison of the CarSim simulation to LPG results for an understeer test. This ﬁgure shows a very good match up to about 0.6g. On this ﬁgure, the Ackerman angle has been subtracted in order to make the data pass through zero. The

Ackerman angle is deﬁned as L/R where L is the wheelbase of the vehicle and R is the radius of the turn. Using the results of the two tests described in this and the previous section and comparing them to the to LPG tests, the CarSim model was considered validated for the Mini Cooper. A more thorough analysis which included the case of gross vehicle weight was done in an internal report to Michelin, BMW, and Clemson [6].

Figure 2.2: Frequency Response Functions (LPG Test vs. CarSim Simulation)

14 Figure 2.3: Spider Chart Comparing CarSim Simulation vs. LPG Test

Figure 2.4: Understeer Test Comparing CarSim Simulation vs. LPG Test

15 Chapter 3

Closed-Loop Modeling and Metrics

3.1 Introduction

This section presents the driver model that is used in this thesis. Following the driver model is an explanation of the International Standards Organization (ISO) double lane change course [1] and the desired trajectory for the vehicle. The next section describes quadratic cost functions as metrics for measuring performance through the lane change.

Using the driver model described, parameters for the driver model were selected using the quadratic cost functions. Finally, as a means to change the vehicle’s handling metrics, hypothetical tires are created and their characteristics are described.

3.2 Driver Model A

In order to negotiate an ISO double lane change, implementing a driver model is necessary. The three driver models examined are constant speed and path-following models which only control steering wheel angle. The driver model chosen for the subsequent analyses will be presented in this section while the remaining two driver models [15, 16] are presented in Appendix B. All of these driver models are relatively simple compared to others described in the literature. However, the goal of this thesis is not to conduct an intensive study of driver models but rather ﬁnd to a driver model that adequately completes

16 the lane change and removes driver variability from the analyses.

In describing a suitable simple driver controller, Genta [15] says the driver model needs to be able to accept a small number of inputs, interpret them and then give an appropriate output. In addition, Genta states the driver is assumed to work oﬀ a reaction time delay and a neuromuscular lag. The reaction time delay takes into account the driver seeing his environment and formulating a response. The neuromuscular lag is the time it takes for the command from the brain to reach the relevant muscles and to perform the required action.

Genta then proposes Equation 3.1 for the driver model. This driver model is not unique to Genta and can be found in several other sources including one by Sutton [21].

Genta says that the time delay and lag fall into two categories–one for common drivers and one for professional drivers. In this thesis only common drivers will be considered.

−τs δ (s) (1 + TLs)e = Kd (3.1) U (s) 1 + TDs

In Equation 3.1 and Figure 3.1, TL is the lead time given to the driver, TD is the neuromuscular lag, Kd is the driver gain, L is the preview distance, τ is the reaction time delay, Yerror is the diﬀerence between the desired and the actual path, ψerror is the diﬀerence between the desired heading angle and the actual heading angle, δ(s) is the steering angle provided by the driver, and U(s) is the input into the driver model.

A general block diagram of the driver/vehicle combination using the transfer function in Equation 3.1 is shown in Figure 3.1. In Appendix B.1, the Simulink block diagram shows the implementation of the driver model incorporated with the CarSim vehicle model.

The beneﬁts of this model are its relative simplicity and ease in implementing. The drawback is that at high speed the driver is very slow to react. Since this model is relatively simple yet provides all the necessary parameters needed for a suitable driver, it is the driver model chosen for the analyses.

17 Figure 3.1: Genta driver and vehicle closed loop system

3.3 ISO Double Lane Change Course and Performance Met-

rics

In this section, the ISO double lane change course boundaries and the desired trajectory are presented. In addition, quadratic cost functions are used as a means for objectively determining performance through the lane change.

3.3.1 ISO Double Lane Change Course

In this thesis the ISO double lane change as shown in Figure 3.2 will be used as the closed-loop test. There are five sections to the course: the first has the car stay in the original lane for 15 meters; the second has the car move laterally 3.5 meters in a 30 meter distance; the third has the vehicle stay in the lane for 25 meters; the fourth has the vehicle return to the original lane in 25 meters; and the fifth is to stay in the original lane for 30 meters. For sections one, three, and five the widths are defined as 1.1T + 0.25, 1.2T + 0.25, and 1.3T + 0.25, respectively, where T is the track width of the vehicle, in meters. For sections two and four, the vehicle does not have to remain within any lateral boundaries.

For simplicity, it is assumed that the trajectory the driver attempts to maintain throughout the course is in the middle of the various segments. A skilled driver would probably take a diﬀerent path that would be much closer to the lane boundaries in sections one, three, and

ﬁve. The desired lateral position and the heading angle deﬁned by Genta [15] are used in this thesis. The lateral position is given by Equation 3.2 and the heading angle is given by

18 Equation 3.3. X and Y are in units of meters.

   Y = 0 for X < 15    Y = 3.5 1 − cos π (X − 15) for 15 ≤ X < 45  2 30  Y = 3.5 for 45 ≤ X < 70 (3.2)    3.5 π  Y = 2 1 + cos 25 (X − 70) for 70 ≤ X < 95    Y = 0 for 95 ≤ X < 125

   ψ0 = 0 for X < 15   n o  ψ = arctan 3.5π sin π (X − 15) for 15 ≤ X < 45  0 60 30  ψ0 = 0 for 45 ≤ X < 70 (3.3)   n o  3.5π π  ψ0 = −arctan 50 sin 25 (X − 70) for 70 ≤ X < 95    ψ0 = 0 for 95 ≤ X < 125

19 Figure 3.2: ISO Lane Change Course

3.3.2 Quadratic Cost Functions

Performance through the double lane change needs to be measured in a way that considers the vehicle’s ability to follow the desired path together with the workload put upon the driver [16]. The workload on the driver includes both mental and physical compo- nents. These performance parameters can be expressed as quadratic integral cost functions computed through the test maneuver. The metric for lateral deviation from the desired

20 path is deﬁned by Equation 3.4.

tf tf J = (y − y)2 dt = y2dt (3.4) 1 ˆ c ˆ e 0 0 where J1 is the total quadratic cost, yc is the y-coordinate of desired C.G. path, y is the actual C.G. path, ye is the instantaneous lateral error, and tf is the ﬁnal time of the observation. Each of the lateral positions (yc, y, and ye) are with respect to a global (ﬁxed) coordinate system. The metric for physical workload can be considered using the cost function in Equation 3.5 as suggested by Horiuchi and Yuhara [16].

tf J = δ2dt (3.5) 2 ˆ 0 where is δ is the steering wheel angle. For the mental workload, Horiuchi and Yuhara suggest using the sum of the leads that their driver model (presented in Appendix B) requires. Equation 3.6 shows how they calculate mental workload.

J3 = TLy + TLψ (3.6)

However, since this analysis does not use their driver model another method for measuring mental workload had to be adopted. This was found in a thesis by Oscarsson

[17] which presents a metric for mental workload as shown in Equation 3.7.

tf J = δ˙2dt (3.7) 3 ˆ 0 Note that the above three quadratic measures are integrated over the entire maneuver and, as such, provide insight into how the vehicle performs for the whole duration.

The objective is to minimize these cost functions through the maneuver under considera- tion. Minimizing J1 means maintaining good path-following ability. Minimizing J2 and J3 implies that the driver workload would be minimal. Each of these metrics in the body of

21 this thesis will be presented in normalized form to the OE vehicle. The nominal values are shown in Appendix D in the interest of completeness. The normalized values allow for an easy comparison to a baseline vehicle, in this case the OE vehicle.

3.4 Driver Parameter Selection

In order to find a relationship between open-loop and closed-loop handling, it is necessary to remove driver variability from the closed-loop system. This necessitates finding a driver model that is the same for all vehicle configurations tested. It is desired for this driver model to be the best possible through the double lane change as measured by weighted quadratic cost functions that were presented in Section 3.3.2. The methodology and selected driver model parameters will be presented in this section and the driver parameters will be used throughout the remaining analyses.

Driver Model A contains the following driver parameters: gain (Kd), lead (TL), lag

(TD), time delay (τ), and preview distance (L). The time delay and lag can be grouped into two categories–those for professional and those for common drivers. For all drivers, these values are independent of task and considered constant [15, 16]. The analysis in this thesis will only consider common drivers. Horiuchi and Yuhara suggest a value of 0.1 seconds for both the lag and time delay terms. The next parameter to consider is the preview distance which is greatly aﬀected by speed and driving task. Horiuchi and Yuhara [16] suggest a preview time of 1.5 seconds which is easily converted to distance based on the speed of the maneuver.

All but two parameters have now been determined–gain (Kd) and lead (TL). The methodology chosen for selecting these two parameters was to use the quadratic cost functions deﬁned above and determine the values of Kd and TL which gave the lowest overall cost. First, weights are assigned to J1, J2 , and J3 in order to put more or less emphasis

22 on the individual metrics. Then a total cost is computed as shown in Equation 3.8

Jtotal = q1J1 + q2J2 + q3J3 (3.8)

where q1, q2, and q3 are weighing factors. In this exercise, the path-following ability is given the highest weight because it is considered very important for the vehicle to make it through the double lane change without violating the course boundaries. The driver physical and mental workloads are given lower priority because of the length of the maneuver. The values q1 = 0.75, q2 = 0.125, and q3 = 0.125 were selected for the analysis that follows, where the individual cost functions J1, J2 , and J3 have been normalized. The weightings will only be used for selecting appropriate driver model parameters. In Chapter 4 all J parameters will be presented individually without a weighted sum.

To select driver gain (Kd) and lead (TL), the simulation was ﬁrst run at a speed of 70 kph using a range of gains and leads. The values of gain and lead that gave the lowest total weighted cost (lowest Jtotal) were then used to determine maximum speed through the lane change without violating the course boundaries. If the maximum speed did not correspond to the speed at which the driver parameters were selected then the speed was increased and the process repeated. The speed at which this occurs is 78 kph with a driver gain of Kd = 0.75 and a lead of TL = 0.2.

Table 3.1 shows the computed cost J1 for a range of Kd and TL settings through an ISO double lane change maneuver at 78 kph. The cost has been normalized with respect to the minimum raw cost, which was found to be at gain Kd = 0.7 and at lead TL = 0.4. These results were for the vehicle with the OE tire. Figures 3.3, 3.4, and 3.5 show the time history data for lateral deviation error, steering wheel angle, and steering wheel angle rate, respectively, at each J’s lowest raw cost. The J values normalized to the lowest values are given as functions of Kd and TL in Tables 3.1, 3.2, and 3.3. Tables 3.2 and 3.3 show similar information to Table 3.1 except they are for physical workload (J2) and mental workload (J3). Table 3.2 shows that the lowest raw cost for

23 Table 3.1: J1 at 78 kph Normalized to Lowest Value Kd

TL 0.55 0.60 0.65 0.7 0.75 0.8 0.85 0.1 3.10 2.82 2.63 2.45 2.26 2.12 2.01 0.2 2.05 1.80 1.60 1.45 1.33 1.26 1.26 0.3 1.71 1.54 1.43 1.33 1.24 1.15 1.15 0.4 1.51 1.31 1.13 1.00 1.22 2.26 1045.47 0.5 1.26 1.15 1.08 4.80 220.44 476.27 20.38

Figure 3.3: Yerror vs. Time for each J’s Lowest Cost

24 Table 3.2: J2 at 78 kph Normalized to Lowest Value Kd

TL 0.55 0.60 0.65 0.7 0.75 0.8 0.85 0.1 1.28 1.46 1.64 1.82 2.01 2.18 2.45 0.2 1.00 1.12 1.24 1.40 1.59 1.84 2.16 0.3 1.02 1.16 1.31 1.49 1.71 2.13 2.94 0.4 1.18 1.46 2.15 3.83 5.90 7.80 94.09 0.5 2.42 4.32 6.00 21.63 93.11 103.27 87.00

Table 3.3: J3 at 78 kph Normalized to Lowest Value Kd

TL 0.55 0.60 0.65 0.7 0.75 0.8 0.85 0.1 1.00 1.19 1.39 1.61 1.88 2.15 2.62 0.2 1.14 1.38 1.66 2.07 2.57 3.29 4.25 0.3 1.80 2.24 2.77 3.53 4.62 6.84 11.15 0.4 3.17 4.80 9.18 19.42 27.10 30.14 25.72 0.5 12.04 23.01 28.47 33.45 27.48 26.54 29.92

physical workload is at TL = 0.2 and Kd = 0.55 which is expected. For any given lead

(TL), it can be seen that as driver gain (Kd) increases, the physical workload of the driver (disregarding the unstable cases) also increases. One can conclude that driver gain relates to physical workload on the driver.

Table 3.3 shows the lowest raw cost for mental workload is found for TL = 0.1 and

Kd = 0.55. Increasing both gain and lead raises a driver’s mental workload which results in the conclusion that these two parameters are related to J3.

Figure 3.6 shows a surface plot of Jtotal with the minimum total cost indicated by the data point which corresponds to TL = 0.2 and Kd = 0.75. Table 3.4 contains the same information as Figure 3.6 but in tabular form for easy viewing of values rather than trends.

25 Figure 3.4: SWA vs. Time for each J’s Lowest Cost

26 Figure 3.5: SWA Rate vs. Time for each J’s Lowest Cost

27 Table 3.4: Jtotal for OE Tire at 78 kph Kd

TL 0.55 0.60 0.65 0.7 0.75 0.8 0.85 0.1 2.61 2.45 2.35 2.27 2.18 2.13 2.14 0.2 1.80 1.66 1.56 1.52 1.51 1.59 1.74 0.3 1.63 1.58 1.58 1.62 1.72 1.99 2.62 0.4 1.68 1.76 2.27 3.66 5.04 6.44 799.08 0.5 2.76 4.28 5.12 10.49 180.41 373.43 29.90

Figure 3.6: Jtotal for OE Tire at 78 kph

The driver parameters selected in this analysis will be used throughout the rest of the document. All relevant driver parameters are in Table 3.5.

Table 3.5: Driver Parameters for Driver Model A τ TD TL Kd L (sec) Parameter 0.1 0.1 0.2 0.75 1.5

28 3.5 Selection of Tire Characteristics

Michelin has provided ﬂat-track testing results for the OE Mini Cooper tire; however, at the time this thesis was written, TWEELs™ for the Mini had not yet been fabricated and, therefore, no data exists. Michelin did provide Pacejka coeﬃcients for an experimental prototype of a TWEEL™ for a BMW 3 Series car. These data were used in the Magic

Formula [3] to generate lateral force and aligning moment versus slip angle curves. Com- bining the Pacejka data for the BMW 3 Series with comments from Michelin [10] led to the creation of the three hypothetical TWEELs™ for the Mini. The eﬀect modeled is that of the tires becoming more or less sensitive to vertical load.

One of the TWEELs™ is supposed to maintain the same total (front + rear) cornering stiﬀness as the OE tire but changes the split (front/rear) of the cornering stiﬀness.

The other two TWEELs™ also change the split as well as raise the total cornering stiﬀness.

The methodology by which the three TWEELs™ were created is discussed in more detail in Appendix C.

Using the method explained in Appendix C to create the TWEELs™, Table 3.6 shows the “garage” values of front and rear, total, and the front to rear split of the cornering stiﬀness. These values for front and rear cornering stiﬀness are dependent on the prevailing weight distribution of the vehicle (OE F/R=60/40).

Table 3.6: Cornering Stiﬀness for each TWEEL™ Cαf Cαr Total Cα (f + r) Front to Rear (N/deg) (N/deg) (N/deg) Split (Cαf/Cαr) OE Tire 1323 948 2271 1.40 TWEEL™ A 1247 1030 2278 1.21 TWEEL™ B 1446 1119 2565 1.29 TWEEL™ C 1561 1327 2888 1.18

Figure 3.7 shows the cornering stiﬀness versus load plot for the candidate TWEELs™.

In this plot the vertical lines mark the garage value loads (at curb + driver loading, C+D) for front and rear wheels which are 3900 and 2600 Newtons, respectively. For the OE tire,

29 Figure 3.7: Cornering Stiﬀness versus Vertical Load for TWEELs™ cornering stiﬀness is almost linear over the given range of vertical loads whereas each of the three TWEELs™ is more non-linear above 3000 N. As vertical load is increased past

5000 N on the TWEELs™, cornering stiﬀness remains about the same (TWEEL™ B) or decreases (TWEELs™ A and C).

Lateral force versus slip angle and vertical load curves are shown in Figure 3.8. In

Figure 3.8 each of the curves represents a diﬀerent vertical load on the tire with the lowest vertical load corresponding to the lowest lateral force curve. Similar plots for each of the

TWEELs™ will be omitted due to the fact they are qualitatively the same as the OE tire except for the noted diﬀerence of cornering stiﬀness versus load.

30 Figure 3.8: Lateral Force vs. Slip Angle for OE Tire

31 Chapter 4

Results and Discussion

4.1 Introduction

In this chapter results of the open-loop and closed-loop handling metrics will be examined for different configurations of the Mini Cooper. First, results will be presented for both open-loop and closed-loop performance by changing the tires using the three hypothetical TWEELs™. Next, these same open-loop and closed-loop metrics will be presented for different weight distributions of the vehicle. Each of these vehicle configurations is considered a realistic change and therefore dramatic improvement or degradation in results is not expected. Finally, combining the TWEELs™ with the weight distribution cases yields twelve configurations which are used to find relationships between open-loop and closed-loop handling.

4.2 Open-Loop and Closed-Loop Performance with Chang-

ing Tires

Shown in Figure 4.1 are the transfer functions for yaw rate and lateral acceleration response of the vehicle. In the upper left-hand plot there are two things to note. First, steady state yaw rate gain is about the same for all tires except TWEEL™ B. Second, yaw

32 Figure 4.1: Frequency Response Functions Comparing OE Tire to TWEELs™ rate bandwidth is much larger for TWEEL™ C than the others. In addition, in the lower right-hand plot, lateral acceleration phase lag at 1 Hz is slightly less than 40 degrees for

TWEEL™ C and is separated from the other tires at 1 Hz which are clustered around 50 to 55 degrees.

Figure 4.2 shows the spider charts generated by simulating the vehicle models through swept-sine steer (open-loop) maneuvers. The transfer functions for yaw rate and lateral acceleration were created and the four parameters were extracted as described in

Chapter 2. Figure 4.2 clearly shows the trends in the open-loop characteristics as the vehicle’s tire sets are changed. When the spider plot shifts to the upper right, the vehicle

33 Figure 4.2: Spider Plot of Open-Loop Handling Metrics with Diﬀerent Tires exhibits more understeering behavior and, conversely, when it shifts to the lower left, the vehicle exhibits less understeering behavior. It can be noted also that the increased total cornering stiﬀness of TWEELs™ B and C (See Table 3.6 above) gives larger areas in the spider plot showing the greater handling potential of the vehicle [2], as should be expected.

The larger front to rear split in cornering stiffness for TWEEL™ A compared to the OE tire accounts for the larger yaw rate bandwidth and smaller lateral acceleration phase lag at 1 Hz for TWEEL™ A. It can also be seen from Figure 4.2 that steady state yaw rate gain and yaw rate damping ratio vary only a small amount for the various configurations compared to the other two parameters on the chart. Table 4.1 contains all of the data presented in Figure 4.2 except in tabular form and also includes the understeer gradient for each configuration.

To evaluate the closed-loop handling, the double lane change performance at the maximum speed (i.e., the maximum speed at which the lane change is traversed without violating lane boundaries) for the car and driver combination is presented. Figure 4.3 shows the course taken by the vehicle with the OE Tire at 78 kph. The desired path has the vehicle

34 Table 4.1: Spider Chart Data for TWEELs™ Lateral SS Yaw Yaw Rate Yaw Rate Acceleration Kus Rate Gain Bandwidth Damping Phase Lag @ 1 (deg/g) (1/s) (Hz) Ratio Hz (deg) OE Tire 0.42 2.35 0.48 -55.84 17.8 TWEEL™ A 0.41 2.52 0.48 -51.88 17.6 TWEEL™ B 0.45 2.61 0.49 -48.57 20.8 TWEEL™ C 0.42 3.10 0.47 -39.68 23.4

Table 4.2: Quadratic Cost Functions at Each Conﬁguration’s Maximum Speed (F/R=60/40) Max Speed (kph) Normalized J 1 Normalized J 2 Normalized J 3 OE Tire 78 1.00 1.00 1.00 TWEEL™ A 77 1.03 1.01 0.96 TWEEL™ B 78 0.92 0.97 0.98 TWEEL™ C 78 0.91 0.90 0.84 staying in the middle of the lanes throughout the course. Figures 4.4 to 4.6 show various vehicle parameters versus time and distance.

Table 4.2 lists each individual J metric (of closed-loop lane change performance) and each vehicle’s maximum speed with which it could successfully negotiate the double lane change maneuver described above without violating the course boundaries. Each J metric is normalized to the equivalent metric computed for the OE tire although TWEEL™ A is normalized at a slightly lower speed than the other three. Since these are cost functions, the goal is to minimize them as much as possible and therefore a value less than 1.00 is considered an improvement over the OE Tire. TWEEL™ C gives between a 9 and 15% improvement in each J-parameter as compared to the OE tire. TWEELs™ A, B, and C each show an improvement in J 3 and TWEELs™ B and C show improvements in both J 2 and J 3.

Using Figure 4.2 and Table 4.2 together one observes that the normalized J 2 and

J 3 appear to decrease as lateral acceleration phase lag at 1 Hz decreases and as yaw rate bandwidth increases. Normalized J 1 shows this same trend except for the vehicle with TWEEL™ A.

35 Figure 4.3: Vehicle Trajectory with OE Tire at Maximum Speed of 78 kph

36 Figure 4.4: Vehicle Response Parameters vs. Time at 78 kph

37 Figure 4.5: Lateral Acceleration vs. Time and Distance at 78 kph

38 Figure 4.6: Lateral Position and Heading Angle vs. Time at 78 kph

39 Table 4.3: Spider Chart Data for Changing Weight Distribution Lateral Weight SS Yaw Yaw Rate Yaw Rate Acceleration Kus Dist. Rate Gain Bandwidth Damping Phase Lag @ 1 (deg/g) (F/R) (1/s) (Hz) Ratio Hz (deg) 60/40, 0.42 2.35 0.48 -55.84 17.8 (OE Tire) 55/45, 0.47 2.27 0.49 -61.53 13.2 (OE Tire) 50/50, 0.51 2.12 0.49 -67.72 9.4 (OE Tire)

4.3 Open-Loop and Closed-Loop Performance Metrics with

Changing Weight Distribution

It was desired to change another ﬁrst-order eﬀect on handling dynamics, namely the static weight distribution, to see if further conclusions can be drawn between open-loop and closed-loop performance metrics. The nominal weight distribution of the vehicle is

60/40 (front to rear). Two other conﬁgurations with front to rear distributions of 55/45 and 50/50 were simulated considering the OE tire set for both conﬁgurations. It is noted that only weights carried by the axles were changed, not the yaw inertias of the vehicle.

Figure 4.7 shows the yaw rate and lateral acceleration transfer functions for the changing weight distributions. From Figure 4.7 there are two important points to note–the large diﬀerences between steady state yaw rate gain for each of the cases in the upper left-hand plot and, in the lower left-hand plot, the lateral acceleration phase lag at 1 Hz.

Figure 4.8 is a spider chart of the simulated vehicle through a swept sine test where the weight distribution of the vehicle was changed. As the C.G. of the vehicle moves rearward, the vehicle has greater steady state yaw rate gain and larger lateral acceleration phase lag at 1 Hz (Figure 4.7). The shift of the rhombi to the lower left with the C.G. moving to the rear also indicates that the vehicle tends toward less understeer. Table 4.3 shows the data in Figure 4.8 in tabular form along with the understeer gradient for each conﬁguration.

40 Figure 4.7: Frequency Response Functions Comparing Weight Distributions

41 Figure 4.8: Spider Plot of Open-Loop Handling Metrics for Changing Weight Distribution

Table 4.4: Quadratic Cost Functions at Each Conﬁguration’s Maximum Speed Max Speed Weight Dist. (F/R) Normalized J Normalized J Normalized J (kph) 1 2 3 60/40, (OE Tire) 78 1.00 1.00 1.00 55/45, (OE Tire) 78 0.98 0.96 1.02 50/50, (OE Tire) 77 0.86 1.60 2.03

Table 4.4 contains the quadratic cost functions normalized to the nominal weight distribution at each configuration’s maximum speed. The configuration that stands out the most is the 50/50 weight distribution. It shows that the driver expends much more physical and mental effort (larger J2 and J3) to complete the lane change in this configuration while tracking the course (J 1) somewhat better than in the nominal configuration albeit at a slightly lower speed.

42 Table 4.5: Legend for Figures in Current Section Conﬁguration (Weight Conﬁguration (Weight Marker Marker Distribution, Tire) Distribution, Tire)

60/40, OE Tire 55/45, TWEEL™ B

60/40, TWEEL™ A 55/45, TWEEL™ C

60/40, TWEEL™ B 50/50, OE Tire

60/40, TWEEL™ C 50/50, TWEEL™ A

55/45, OE Tire 50/50, TWEEL™ B

55/45, TWEEL™ A 50/50, TWEEL™ C

4.4 Qualitative Analysis Relating Open and Closed-Loop Han-

dling Metrics

In this section, it is desired to present additional qualitative comparisons between the four-parameter open-loop handling measures and the closed-loop quadratic performance measures by combining the considerations in the previous two sections to cover more con-

figurations of the vehicle. Therefore each tire is combined with each weight distribution yielding twelve configurations. The results are presented in Figures 4.9, 4.10, 4.11, and 4.12 with Table 4.5 listing the symbols used in the figures. To remove speed as a variable in this analysis, all configurations were simulated at a speed of 77 kph, the maximum speed with which all configurations could successfully complete the lane change.

First, the upper two plots in Figure 4.9 show the lateral acceleration phase lag at 1 Hz and yaw rate bandwidth compared to the physical workload as measured by J2. These plots show that physical workload is reduced as yaw rate bandwidth increases and lateral acceleration phase lag at 1 Hz decreases. The lower plots of Figure 4.9 show yaw rate bandwidth and lateral acceleration phase lag at 1 Hz compared to the mental workload (J3). The plots clearly show that mental workload is reduced as yaw rate bandwidth increases and lateral acceleration phase lag at 1 Hz decreases.

43 Figure 4.9: Relating Select Open and Closed-Loop Metrics in 2-D Plots

44 Figure 4.10: 3-D Plot Comparing SS Yaw Rate Gain and Yaw Rate Bandwidth to J1

Figure 4.10 is a 3-D plot showing the observed relationship between steady state yaw rate gain and yaw rate bandwidth versus the metric for lateral deviation from the desired path (J1). The points stretch from smaller yaw rate bandwidth and steady state yaw rate gain and extend towards greater yaw rate bandwidth and steady state yaw rate gain. As should be expected, the plot suggests that as steady state yaw rate gain and yaw rate bandwidth increase, the metric J1 is reduced, indicating improved tracking performance. Figures 4.11 and 4.12 show lateral acceleration phase lag at 1 Hz and yaw rate bandwidth versus J2 and J3, respectively. This is the same information as presented in Figure 4.9 but is shown in three dimensional space to get a better understanding of how mental

(J2) and physical (J3) workload correspond with both parameters. The same conclusion is drawn– as lateral acceleration phase lag at 1 Hz decreases and yaw rate bandwidth increases

45 Figure 4.11: 3-D Plot Comparing Lateral Accel. Phase Lag and Yaw Rate Bandwidth to J2 physical and mental workload are reduced.

It can also be observed that the more aggressive design, TWEEL™ C (in terms of highest overall cornering stiffness), consistently gives lower J1, J2, and J3 costs regardless of the weight distribution. However, in some cases, the particular vehicle characteristic that was changed does have an effect on whether specific relationships hold. For example, the

50/50 OE tire case shows larger yaw rate bandwidth and lower lateral acceleration phase lag at 1 Hz than the 50/50 TWEEL A, but it also gives the largest J2 and J3 costs. Other pairs of performance metrics were also considered (e.g. steady state yaw rate gain and yaw rate bandwidth versus mental workload), but no clear qualitative relationships were detected and therefore those results are not presented.

46 Figure 4.12: 3-D Plot Comparing Lateral Accel. Phase Lag and Yaw Rate Bandwidth to J3

47 Chapter 5

Conclusions and Future Work

5.1 Conclusions

In this thesis, it was attempted to establish relationships between traditional open- loop handling metrics and quadratic cost functions measuring closed-loop performance. To this end, a vehicle model was built for the Mini Cooper and was validated with track testing data. In addition, a driver model that included lead, lag, time delay, and preview distance parameters was constructed. Quadratic cost functions were then adopted to take into account lane change tracking performance and driver mental and physical workload.

Driver parameters were then selected at a given speed using the quadratic cost functions to minimize the total cost with high priority given to successfully completing the lane change. Mental and physical workloads were given less priority due to the short length of the maneuver. The selected driver parameters were then used for the remaining analyses to determine relationships between open-loop and closed-loop performance metrics.

TWEEL™ characteristics were then presented in a cornering stiffness versus load plot to show the load sensitivity of the TWEEL™. The TWEELs™ and changing the weight distribution of the vehicle were used as a means of changing the vehicle configuration. The different vehicle configurations were used to make a qualitative analysis between the four- parameter (open-loop) handling performance metrics and double lane change quadratic

48 metrics. The following clear relationships exist:

• As steady state yaw rate gain and yaw rate bandwidth increased, path-following ability

(J1) is generally enhanced.

• As yaw rate bandwidth increases and lateral acceleration phase lag at 1 Hz decreases,

both physical (J2) and mental (J3) workload on the driver are reduced.

Relationships among the other performance metrics were found to be less definitive with the approach taken here and the vehicle (and its configurations) considered in the present work. It was also noted that even for the specific relationships identified, the extent of the correlation could be a function of the vehicle design characteristics (cornering stiffness or weight distribution) that were changed to extract the relationships.

5.2 Future Work

Future work in this area should consider many different configurations for many different vehicles to give a large array of data upon which to make statistical correlations.

The diﬀerent vehicles should come from a variety of diﬀerent classes including large and small sedans, sports cars, SUVs, and trucks. Given the current data, the car at a loading corresponding to gross vehicle weight could be included in the correlation to double the number of data points.

In addition, a more sophisticated method of correlation could be possible with more data points. In order to validate closed-loop performance to a baseline vehicle, testing needs to be conducted measuring the quadratic cost functions described in this thesis. A more complex driver model could be coupled to the vehicle model in order to better emulate driver behavior.

49 APPENDICES

50 Appendix A

Vehicle Model Parameters

Some of the documents referenced in this appendix are internal documents to Miche- lin, BMW, and Clemson University and are not, therefore, available to any other parties.

Vehicle yaw inertia was not measured as part of K&C testing. However, once the model was built with all other parameters known, yaw inertia was changed until a good match was achieved between LPG testing and simulation. In the case for curb plus driver a value of 1750 kg − m2 was found to match.

51 A.1 Vehicle Properties

Table A.1: Vehicle Properties Description Curb + Driver Units Source

Total vehicle mass 1323 kg [20] Page 1

Wheelbase 2468 mm [4] Figure 2

Track width (front) 1453 mm [4] Figure 2

Track width (rear) 1475 mm [4] Figure 2

CG location longitudinally 981 mm [22] Table 2 (distance from front axle – a)

CG location longitudinally 1487 mm [22] Table 2 (distance from rear axle – b)

CG location vertically 517 mm [22]Table 2

52 A.2 Suspension Parameters

Table A.2: Suspension Parameters Description Curb + Driver Units Source

Front spring force vs. Spring Rate N/mm [4] Section ZF Figure Compression Curve Z1

Rear spring force vs. Spring Rate N/mm [4] Section ZR Figure Compression Curve Z1

Diﬀerence between overall roll [4] Vehicle summary,

stiﬀness and that provided by 577 N-m/deg Roll test, Table 4 &

springs (front) Table 41

Diﬀerence between overall roll [4] Vehicle summary,

stiﬀness and that provided by 422 N-m/deg Roll test, Table 4 &

springs (rear) Table 41

Front shock travel rate vs. Front Shock N-s/mm [4] Section A Damping force Curve

Rear shock travel rate vs. Rear Shock N-s/mm [4] Section A Damping force Curve

53 A.3 Suspension Kinematic Parameters

Table A.3: Left Front Suspension Kinematic Parameters Description Curb + Driver Units Source

Left Front

[23]Inertial

Parameters and CG

Wheel spin moment of inertia 1.637 kg − m2 locations, values are

scaled using rolling

radius

[4] Table 1A Loaded Static toe left front 0.125 deg Geometry Summary

[4] Table 1A Loaded Static camber left front -0.28 deg Geometry Summary

Camber Curve [4] Section ZF Figure Camber change due to jounce deg Sheet Z3

Lateral Lateral displacement change [4] Section ZF Figure Displacement mm due to jounce Z6 Sheet

Toe Change [4] Section ZF Figure Toe steer due to jounce deg Sheet Z2

Caster Curve [4] Section ZF Figure Caster change due to jounce deg Sheet Z8

Longitudinal Longitudinal movement of [4] Section ZF Figure Movement mm wheel due to jounce Z7 Sheet

54 Table A.4: Right Front Suspension Kinematic Parameters Description Curb + Driver Units Source

Right Front

[23] Inertial

Parameters and CG

Wheel spin moment of inertia 1.637 kg − m2 locations, values are

scaled using rolling

radius

[4] Table 1A Loaded Static toe right front 0.125 deg Geometry Summary

[4] Table 1A Loaded Static camber right front -0.62 deg Geometry Summary

Camber Curve [4] Section ZF Figure Camber change due to jounce deg Sheet Z3

Lateral Lateral displacement change [4] Section ZF Figure Displacement mm due to jounce Z6 Sheet

Toe Change [4] Section ZF Figure Toe steer due to jounce deg Sheet Z2

Caster Curve [4] Section ZF Figure Caster change due to jounce deg Sheet Z8

Longitudinal Longitudinal movement of [4] Section ZF Figure Movement mm wheel due to jounce Z7 Sheet

55 Table A.5: Left Rear Suspension Kinematic Parameters Description Curb + Driver Units Source

Left Rear

[23] Inertial

Parameters and CG

Wheel spin moment of inertia 1.376 kg − m2 locations, values are

scaled using rolling

radius

[4] Table 1A Loaded Static toe left rear 0.175 deg Geometry Summary

[4] Table 1A Loaded Static camber left rear -1.7 deg Geometry Summary

Camber Curve [4] Section ZR Figure Camber change due to jounce deg Sheet Z13

Lateral Lateral displacement change [4] Section ZR Figure Displacement mm due to jounce Z16 Sheet

Toe Change [4] Section ZR Figure Toe steer due to jounce deg Sheet Z12

Longitudinal Longitudinal movement of [4] Section ZR Figure Movement mm wheel due to jounce Z17 Sheet

56 Table A.6: Right Rear Suspension Kinematic Parameters Description Curb + Driver Units Source

Right Rear

[23] Inertial

Parameters and CG

Wheel spin moment of inertia 1.376 kg − m2 locations, values are

scaled using rolling

radius

[4] Table 1A Loaded Static toe right rear 0.175 deg Geometry Summary

[4] Table 1A Loaded Static camber right rear -1.68 deg Geometry Summary

Camber Curve [4] Section ZR Figure Camber change due to jounce deg Sheet Z13

Lateral Lateral displacement change [4] Section ZR Figure Displacement mm due to jounce Z16 Sheet

Toe Change [4] Section ZR Figure Toe steer due to jounce deg Sheet Z12

Longitudinal Longitudinal movement of [4] Section ZR Figure Movement mm wheel due to jounce Z17 Sheet

57 A.4 Compliance Parameters

Table A.7: Left Front Compliance Parameters Description Curb + Driver Units Source

Left Front

Coeﬃcient of change in toe [4] Section XF, List 6.30E-05 deg/N per change in Fx XF

Coeﬃcient of change in steer [4] Section YF, List -4.00E-05 deg/N angle per change in Fy YF

Coeﬃcient of change in steer [4] Section YF, List 4.90E-03 deg/N-m angle per change of tire Mz YF

Coeﬃcient of change in [4] Section YF, List inclination angle per change 1.94E-04 deg/N-m YF in Fy

Coeﬃcient of change in [4] Section YF, List inclination angle per change -6.80E-03 deg/N-m YF in Mz

Coeﬃcient of change in X [4] Section XF, List displacement per change in 1.84E-03 mm/N XF Fx

Coeﬃcient of change in Y [4] Section YF, List 2.41E-04 mm/N displacement per change in Fy YF

58 Table A.8: Right Front Compliance Parameters Description Curb + Driver Units Source

Right Front

Coeﬃcient of change in toe [4] Section XF, List 8.00E-05 deg/N per change in Fx XF

Coeﬃcient of change in steer [4] Section YF, List -3.10E-05 deg/N angle per change in Fy YF

Coeﬃcient of change in steer [4] Section YF, List 4.83E-03 deg/N-m angle per change of tire Mz YF

Coeﬃcient of change in [4] Section YF, List inclination angle per change 1.93E-04 deg/N-m YF in Fy

Coeﬃcient of change in [4] Section YF, List inclination angle per change -6.87E-03 deg/N-m YF in Mz

Coeﬃcient of change in X [4] Section XF, List displacement per change in 1.95E-03 mm/N XF Fx

Coeﬃcient of change in Y 2.35E-04 mm/N [4] Section YF List YF displacement per change in Fy

59 Table A.9: Left Rear Compliance Parameters Description Curb + Driver Units Source

Left Rear

Coeﬃcient of change in toe [4] Section XR , List 0.00E+00 deg/N per change in Fx XR

Coeﬃcient of change in steer [4] Section YR , List -2.10E-05 deg/N angle per change in Fy YR

Coeﬃcient of change in steer [4] Section YR , List 2.33E-04 deg/N-m angle per change of tire Mz YR

Coeﬃcient of change in [4] Section YR , List inclination angle per change 2.74E-04 deg/N-m YR in Fy

Coeﬃcient of change in [4] Section YR , List inclination angle per change -9.03E-03 deg/N-m YR in Mz

Coeﬃcient of change in X [4] Section XR , List displacement per change in 0.00E+00 mm/N XR Fx

Coeﬃcient of change in Y [4] Section YR , List 4.84E-04 mm/N displacement per change in Fy YR

60 Table A.10: Right Rear Compliance Parameters Description Curb + Driver Units Source

Right Rear

Coeﬃcient of change in toe [4] Section XR , List 0.00E+00 deg/N per change in Fx XR

Coeﬃcient of change in steer [4] Section YR , List -8.00E-06 deg/N angle per change in Fy YR

Coeﬃcient of change in steer [4] Section YR , List 8.67E-04 deg/N-m angle per change of tire Mz YR

Coeﬃcient of change in [4] Section YR , List inclination angle per change 2.77E-04 deg/N-m YR in Fy

Coeﬃcient of change in [4] Section YR , List inclination angle per change -9.40E-03 deg/N-m YR in Mz

Coeﬃcient of change in X [4] Section XR , List displacement per change in 0.00E+00 mm/N XR Fx

Coeﬃcient of change in Y [4] Section YR , List 4.86E-04 mm/N displacement per change in Fy YR

61 A.5 Tire Parameters

Table A.11: Tire Parameters Description Curb + Driver Units Source

[24] XYZ Rigidity Spring rate 232 N/mm Test

Maximum allowed force 100000 N CarSim Default

CarSim Rolling resistance coeﬃcient – Default

(curve) Slip angle versus Fy [5] Michelin Pacejka data

(curve) Slip angle versus Mz [5] Michelin Pacejka data

Camber thrust coeﬃcient (curve) [5] Michelin versus Fz Pacejka data

[24] XYZ Rigidity Tire lateral relaxation length 368.2 mm Test

62 A.6 Kingpin Geometry

Table A.12: Kingpin Geometry Parameters Description Curb + Driver Units Source

Left Lateral oﬀset @ center 51.16 mm [4] Section S, List S (spindle oﬀset)

Right Lateral oﬀset @ center 51.37 mm [4] Section S, List S (spindle oﬀset)

Left kingpin inclination angle 11.37 deg [4] Section S, List S

Right kingpin inclination 11.39 deg [4] Section S, List S angle

Left X coord. Of KP @ center -6.05 mm [4] Section S, List S (spindle trail)

Right X coord. Of KP @ -7.73 mm [4] Section S, List S center (spindle trail)

Left caster angle 2.81 deg [4] Section S, List S

Right caster angle 3.43 deg [4] Section S, List S

63 Appendix B

Driver Models

B.1 Driver Model A

This model was brieﬂy explained in Section 3.2 with a general block diagram and the key parameters of the model. This section takes a more in-depth look at the actual

MATLAB/Simulink block diagram and how it operates. The general block diagram is once again shown here in Figure B.1. Driver model A uses a look-up table for both commanded lateral position (Y) and heading angle (ψ). Lateral position error is then divided by the preview distance (L) and that quotient and yaw angle error are passed through the driver transfer function which outputs a steer angle so the vehicle dynamics can be calculated.

Figure B.2 contains the actual Simulink block diagram used for analysis. Starting at the “ISO Lane Change Course” both the lateral position error and heading angle error

Figure B.1: Original Genta Driver Model Block Diagram

64 are being used by the driver model to determine the next time step’s output. Then the lateral position is divided by the preview distance, which in the MATLAB code running the Simulink model, is given by a time and converted to a distance based on speed of the maneuver. Then the terms are added together and passed through the driver model which includes the gain (Kd), time delay (τ), lead (TL) and lag (TD) transfer function, then through a rate and a saturation limiter.

The rate limiter block is intended to simulate the rate at which a driver can input steering wheel angle into the vehicle. This value was obtained from Day and Metz [25] and is set to a value of 720 degrees/second for hand wheel angle. However, since this model is using road wheel angle as an input into CarSim, the steering rate (720 deg/sec) is divided by the steering ratio (13.8:1) provided in the K&C report [4]. The saturation block is intended to replicate the physical limits of the steering system by limiting the maximum hand wheel angle to 405 degrees [4] which is then divided by the steering ratio to get the value in terms of road wheel angle. The steer angle is then passed into CarSim as road wheel angle and CarSim then computes all of the vehicle dynamics. The outputs from CarSim are: vehicle heading angle, side slip angle, forward velocity, yaw rate, lateral acceleration, lateral position of the vehicle in the global reference frame, steering wheel torque, and slip angles at each of the tires. Some of these variables are stored in the workspace and used for post-processing in MATLAB while others are used as feedback for the next time step. Then the X-position of the vehicle is calculated based on the forward velocity (Vx), heading angle (ψ), and side slip angle (β) so the model can determine the next commanded Y-position and heading angle. Finally, each of the quadratic cost functions are computed and sent to the workspace for post-processing.

65 Figure B.2: Driver Model A Simulink/CarSim Block Diagram

66 B.2 Driver Model B

This section presents a simpliﬁed version of driver model A presented in Section

3.2. Genta explains that Equation 3.1 can be simpliﬁed by neglecting the lead time and combining all lags and delays into τ [15]. Reducing Equation 3.1 by this method yields

Equation B.1.

δ(s) = K e−τs (B.1) U(s) d

Next, e−τs can be approximated as a power series and truncated to only the linear term, ignoring higher order terms. Rewriting the equation and substituting ψ for U(s) gives

Equation B.2.

˙ τδ (t) + δ (t) = −Kd [ψ (t) − ψ0 (t)] (B.2)

In Equation B.2, τ is the time delay of the driver, δ is the steer angle provided by the driver, Kd is the gain, ψ is the heading angle of the vehicle, and ψ0 is the commanded heading angle.

Finally, Genta proposes adding a term to account for the predictive behavior of the driver. This term is in the form of a distance the driver is looking ahead of his position (i.e., the preview distance). Equation B.3 is the control law used in the Simulink model which is used in co-simulation with CarSim and includes the driver preview distance.

∆y (t) τδ˙ (t) + δ (t) = −K ψ (t) − ψ (t) + (B.3) d 0 L

In Equation B.3, ∆y (t) is the lateral deviation of the C.G. of the vehicle from the prescribed path and L is the preview distance.

The beneﬁt of driver model B is that it is very simple with only a time delay, gain, and look ahead distance. The drawback to this driver model is that the driver does not have any lead which makes the model very slow to react. Because of this drawback the

67 model is considered unacceptable for this thesis.

. Equation B.3 is the control law used in the MATLAB/Simulink model. This differential equation uses the difference between desired heading angle and actual heading angle, the difference between desired lateral position of the C.G. and actual lateral position of the C.G., and the current steering wheel angle as inputs. The output of the control law is steering wheel angle rate which is integrated and passed into CarSim which calculates the vehicle dynamics. Some of the output variables are then used in the next time step for the driver model .

68 Figure B.3: Driver Model B Simulink/CarSim Block Diagram

69 B.3 Driver Model C

Another driver model evaluated in this thesis is one presented by Horiuchi and

Yuhara [16]. As with the model presented in Equation 3.1, this model contains gain, lead, lag, and time delay terms. The authors propose a closed loop system as shown in Figure

B.4. The outer loop “is needed to make the vehicle accurately follow the desired path, while the inner loop that feeds back yaw angles is necessary to give appropriate damping to the closed loop system [16].”

Compared to driver model A, this model has separate gains (Ky and Kψ) and lead terms (TLy and TLψ) for both the lateral position and the heading angle. It also contains a

−τs time delay (e ) and a ﬁrst order lag (T 1) that the authors say are due to central nervous processing time and dynamics of the muscular system. An interesting point that the authors make about the delays is that they are independent of the task the driver is performing, therefore their values are considered constant. According to Horiuchi and Yuhara, the driver will adjust his or her own gains and leads to achieve the desired closed loop system.

The transfer function for the lateral position is given in Equation B.4 below.

ψ(s) = Ky (TLys + 1) (B.4) yerror(s)

The transfer function for the yaw angle is given in Equation B.5 below.

δ(s) e−τs = Kψ (TLψ + 1) (B.5) ψerror(s) T1s + 1

For driver model C, the beneﬁts are the driver model maintains the course very well as compared to the two previous driver models A and B. The drawback for model C is that it has more parameters than the previous two. This makes it more diﬃcult to optimize the driver parameters for the double lane change maneuver. Because of this drawback, this model was not chosen for the analysis in this thesis.

Figure B.5 shows the MATLAB/Simulink model. The model works by inputting

70 the commanded lateral position and ﬁnding the error between that and the actual lateral position which is then divided by the preview distance (L) and passed through the lateral position transfer function. That value is then added to the heading angle error and passed through the heading angle transfer function. As with the driver model A, this model also uses a rate limiter to mimic the rate at which a real driver can input steering wheel angle.

The value of this block is 720 degrees/second [25] divided by the steering ratio (13.8:1) [4].

A saturation block is also included to mimic the physical limits of the steering system which is considered to be 405 degrees of hand wheel angle [4] divided by the steering ratio. This steer angle is then sent to CarSim which computes all of the vehicle dynamics then outputs those variables back into Simulink. Some of those parameters are written to the workspace for post-processing in MATLAB while others are used for the next time step within the model. The next X-position for the model is calculated using the forward velocity (Vx), yaw angle (ψ), and side slip angle (β) which is then sent into the ISO Lane Change course to ﬁnd the next commanded Y-position and heading angle.

71 Figure B.5: Driver Model C Simulink/CarSim Block Diagram

72 Figure B.4: Horiuchi Driver/Vehicle Closed Loop System

73 Appendix C

Methodology for Creating the TWEELs™

One of the methods used to change the vehicle was by fitting it with TWEELs™ but since they do not currently exist for the Mini Cooper, hypothetical ones had to be created based on comments from Michelin [10] and data from the BMW 3 Series TWEEL™. The only characteristic of the TWEEL™ being investigated in this work is the load sensitivity which affects the cornering stiffness versus load plot. Michelin characterizes tire force and moment curves using coefficients which can then be used in Pacejka’s Magic Formula [3].

Pacejka’s equations for lateral force, as Michelin uses them, are presented in Equation C.1

Y = D · sin(C · arctan(B · phi))

E phi = (1 − E) · delta + B arctan(B · delta)

D = (a1 · Z + a2) · Z BCD = a · sin 2 · arctan Z (1 − a · |gamma|) (C.1) 3 a4 5

B = BCD/(C·D)

C = a0

E = a6 · Z + a7

74 where Y is the lateral force in Newtons, delta is the slip angle of the tire in degrees, Z is the vertical load in kilo-Newtons, gamma is the camber angle in degrees, and ai are the coefficients provided by Michelin. Note that this formulation eliminates offsets due to conicity and plysteer at zero slip angle. The part that is of interest for this thesis is BCD which is the cornering stiffness. Assuming camber is zero, it is evident that the cornering stiffness depends on the coefficients a3 , a4, and the vertical load. From the 3 Series

TWEEL™ the coefficients are a3 = 1392.9544 and a4 = 8.9047. Using these coefficients yields a curve similar to what is desired for the Mini but slightly lower overall cornering stiffness than desired. The goal was to create three TWEELs™ with each having one of the following three properties: the same overall cornering stiffness as the OE tire (front + rear) while changing the split (front/rear), increased overall cornering stiffness from the OE tire while changing the split, and raising the overall cornering stiffness even more while bringing the split even closer to one. Another overall goal for each of these TWEELs™ was to make them a realistic change in cornering stiffness that Michelin could achieve and would provide desirable handling characteristics for the vehicle.

Using the BMW 3 Series TWEEL™ as a baseline, a3 and a4 were modified by certain percentages iteratively to find the desired cornering stiffness versus load curve. Equation

C.2 shows how a3 and a4 were modiﬁed to achieve the diﬀerent hypothetical TWEELs™.

It was found in this range of a3 and a4 that a3 controls the overall cornering stiﬀness and that a4 controls the slope of the curve. Using this method yields Figure 3.7.

TWEEL™ A a3 = 1392.9544 × 0.94 a4 = 8.9047 × 0.60

TWEEL™ B a3 = 1392.9544 × 1.20 a4 = 8.9047 × 0.76 (C.2)

TWEEL™ C a3 = 1392.9544 × 1.15 a4 = 8.9047 × 0.55

The aligning moment used for the Mini was the BMW 3 Series TWEEL™ aligning moment because Michelin has not provided much detail of the Mini TWEEL’s™ aligning moment characteristics. It is anticipated that the Mini TWEEL™ will be similar to the 3

75 Series TWEEL™ because the same property that inﬂuences the load sensitivity will govern the aligning moment. The shear band, as shown in Figure ?? is very stiﬀ compared to a standard pneumatic tire and therefore deforms less. This causes a shorter moment arm and therefore produces less aligning moment than a pneumatic tire.

The Pacejka Magic Formula equations which Michelin uses for the aligning moment are shown in Equation C.3

N = D · sin(C · arctan(B · phi))

phi = (1 − E) · delta + E/B arctan(B · delta)

D = (c1 · Z + c2) · Z

−c ·Z BCD = (c3 · Z + c4) · Z · e 5 · (1 − c6 · |gamma|) (C.3)

B = BCD/C·D

C = c0

2 E = c7 · Z +c8 · Z + c9 · (1 − c10 · |gamma|) where N is the aligning moment in Newton-meters, delta is the slip angle of the tire in degrees, Z is the vertical load on the tire in kilo-Newtons, gamma is the camber angle in degrees, and ci are the coeﬃcients provided by Michelin. Using Equation C.3 the aligning moment curves used in CarSim are generated and shown in Figure C.1. In Figure C.1, it should be noted that each curve corresponds to increasing vertical load; therefore as more vertical load is applied more aligning moment is created by the tire.

76 Figure C.1: BMW 3 Series Aligning Moment Used for Mini Cooper in CarSim

77 Appendix D

Open and Closed-Loop Raw Data

In this appendix, the data used in Section 4.4 for drawing relationships between open-loop and closed-loop handling will be presented. Table D.1 shows all of the open-loop handling metrics for each of the twelve cases. Table D.2 contains the raw values for each of the quadratic cost functions. The double lane change was conducted at 77 kph for all cases in order to remove speed as a variable. Table D.3 contains the entire list of normalized quadratic metrics. Each case is normalized to the OE vehicle (tire and weight distribution).

This is the data used to create the relationships presented in Section 4.4.

78 Table D.1: 4 Parameter Data for 12 Conﬁgurations SS Yaw Yaw Rate Yaw Rate Lateral Weight Dist., Tire Rate Gain Bandwidth Damping Acceleration Phase (1/s) (Hz) Ratio Lag @ 1 Hz (deg) 60/40, OE Tire 0.42 2.35 0.48 -55.84 60/40, TWEEL™ A 0.42 2.52 0.48 -51.88 60/40, TWEEL™ B 0.45 2.61 0.49 -48.57 60/40, TWEEL™ C 0.42 3.10 0.47 -39.68 55/45, OE Tire 0.47 2.27 0.49 -61.53 55/45, TWEEL™ A 0.49 2.35 0.49 -60.32 55/45, TWEEL™ B 0.50 2.52 0.49 -54.85 55/45, TWEEL™ C 0.48 2.89 0.48 -47.06 50/50, OE Tire 0.51 2.12 0.49 -67.72 50/50, TWEEL™ A 0.56 2.04 0.49 -69.98 50/50, TWEEL™ B 0.56 2.27 0.50 -62.27 50/50, TWEEL™ C 0.53 2.61 0.49 -54.61

Table D.2: Raw Values of Quadratic Cost Functions at 77 kph 2 2 2 Weight Dist., Tire J1, m · sec J2, rad · sec J3, rad /sec 60/40, OE Tire 1.17 2.98 84.95 60/40, TWEEL™ A 1.26 3.07 85.33 60/40, TWEEL™ B 1.07 2.80 79.68 60/40, TWEEL™ C 1.06 2.66 70.73 55/45, OE Tire 1.14 2.78 83.26 55/45, TWEEL™ A 1.21 2.92 87.97 55/45, TWEEL™ B 1.05 2.61 78.18 55/45, TWEEL™ C 1.01 2.46 69.16 50/50, OE Tire 1.05 4.87 180.07 50/50, TWEEL™ A 1.12 4.05 147.70 50/50, TWEEL™ B 0.98 3.84 152.97 50/50, TWEEL™ C 0.98 2.49 80.44

79 Table D.3: Normalized Values of Quadratic Cost Functions at 77 kph Weight Dist., Tire Normalized J1 Normalized J2 Normalized J3 60/40, OE Tire 1.00 1.00 1.00 60/40, TWEEL™ A 1.08 1.03 1.00 60/40, TWEEL™ B 0.92 0.94 0.94 60/40, TWEEL™ C 0.90 0.89 0.83 55/45, OE Tire 0.97 0.93 0.98 55/45, TWEEL™ A 1.03 0.98 1.04 55/45, TWEEL™ B 0.89 0.88 0.92 55/45, TWEEL™ C 0.86 0.83 0.81 50/50, OE Tire 0.90 1.64 2.12 50/50, TWEEL™ A 0.96 1.36 1.74 50/50, TWEEL™ B 0.84 1.29 1.80 50/50, TWEEL™ C 0.84 0.84 0.95

80 Appendix E

CarSim File Formats

When using CarSim for data analysis, whether it be the Spectrum Analyzer tool or the time history solver, it was necessary to post-process the data in an easy-to-read and presentable manner. To this end, the data from CarSim was imported into MATLAB for post-processing. CarSim uses two file formats for data storage–an ERD header file, developed by the University of Michigan at Ann Arbor, and a .bin file which is a binary

ﬁle that contains all of the data. The data must then be imported into MATLAB so it can be used in a meaningful way. The steps for doing this follow with sample MATLAB code.

This method was developed by Anup Khekare, a fellow graduate student working on the

TWEEL™ project.

• Read the .bin ﬁle into MATLAB - ﬁd = fopen(’FFT133_Run1396.bin’,’r’);

• Decode the binary file - D=fread(fid,’float32’);

At this point the data is contained in one long column with nothing distinguishing the different channels in the data. Therefore it is necessary to break the data into multiple rows of data containing things such as frequency, yaw rate gain, etc. In order to do this two things have to be known–the number of samples per channel and the number of channels in the ERD file. These two things can be found in the ERD file as shown in Figure E.1.

81 Figure E.1: Sample ERD ﬁle

82 The ﬁnal part consists of converting this long column of data into the diﬀerent channels. The MATLAB code is shown below. Once this is done the data can then be post-processed as necessary.

for m=1:110

FFT133_Run1396(:,m)=D(((10*(m-1))+1):(10*m));

end

83 Appendix F

MATLAB Programs

F.1 Model_Validation.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Program for validating CarSim Model to LPG Test Data

% Reads in the five runs at LPG and averages them

% Plots frequency response functions and spider plots

% to compare the model and testing data.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all

close all

clc

%% Read in the data from the CarSim FFT 115 CURB_21_01fid = fopen(’FFT115_CURB_21_1_20.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data treated

% in CARSIM

D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM stores

% binary data in this format.

84 FFT_115_CURB_21_01=zeros(17,90); for m=1:90 % 90 corresponds to the number of samples per channel

FFT_115_CURB_21_01(:,m)=D(((17*(m-1))+1):(17*m));

% 17 corresponds to the number of channels in the ERD file end

%% Read in the data from the CarSim FFT 115 CURB_21_02 fid = fopen(’FFT115_CURB_21_2_20.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data treated

% in CARSIM

D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

FFT_115_CURB_21_02=zeros(17,90); for m=1:90 % 90 corresponds to the number of samples per channel

FFT_115_CURB_21_02(:,m)=D(((17*(m-1))+1):(17*m));

% 17 corresponds to the number of channels in the ERD file end

%% Read in the data from the CarSim FFT 115 CURB_21_03 fid = fopen(’FFT115_CURB_21_3_20.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data treated

% in CARSIM

D=fread(fid,’float32’); % float32 has been used to decode,

% since by default CARSIM stores binary data in this format.

FFT_115_CURB_21_03=zeros(17,90); for m=1:90 % 90 corresponds to the number of samples per channel

85 FFT_115_CURB_21_03(:,m)=D(((17*(m-1))+1):(17*m));

% 17 corresponds to the number of channels in the ERD file end

%% Read in the data from the CarSim FFT 115 CURB_21_04 fid = fopen(’FFT115_CURB_21_4_20.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data

% treated in CARSIM

D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

FFT_115_CURB_21_04=zeros(17,90); for m=1:90 % 90 corresponds to the number of samples per

% channel

FFT_115_CURB_21_04(:,m)=D(((17*(m-1))+1):(17*m));

% 17 corresponds to the number of channels in the ERD file end

%% Read in the data from the CarSim FFT 115 CURB_21_05 fid = fopen(’FFT115_CURB_21_5_20.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data

% treated in CARSIM

D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT_115_CURB_21_05=zeros(17,90); for m=1:90 % 90 corresponds to the number of samples

86 % per channel

FFT_115_CURB_21_05(:,m)=D(((17*(m-1))+1):(17*m));

% 17 corresponds to the number of channels in the ERD file end

%% Average all of the LPG test data

FFT_115_CURB_21_Frequency_Average= (FFT_115_CURB_21_05(1,:)+

FFT_115_CURB_21_04(1,:)+FFT_115_CURB_21_03(1,:)+...

FFT_115_CURB_21_02(1,:)+ FFT_115_CURB_21_01(1,:))/5;

FFT_115_CURB_21_YRGain_Average= (FFT_115_CURB_21_05(9,:)+

FFT_115_CURB_21_04(9,:)+FFT_115_CURB_21_03(9,:)+...

FFT_115_CURB_21_02(9,:)+ FFT_115_CURB_21_01(9,:))/5;

FFT_115_CURB_21_YRPhase_Average= (FFT_115_CURB_21_05(10,:)+

FFT_115_CURB_21_04(10,:)+FFT_115_CURB_21_03(10,:)+...

FFT_115_CURB_21_02(10,:)+ FFT_115_CURB_21_01(10,:))/5;

FFT_115_CURB_21_AyGain_Average= (FFT_115_CURB_21_05(12,:)+

FFT_115_CURB_21_04(12,:)+FFT_115_CURB_21_03(12,:)+...

FFT_115_CURB_21_02(12,:)+ FFT_115_CURB_21_01(12,:))/5;

FFT_115_CURB_21_AyPhase_Average= (FFT_115_CURB_21_05(13,:)+

FFT_115_CURB_21_04(13,:)+FFT_115_CURB_21_03(13,:)+...

FFT_115_CURB_21_02(13,:)+ FFT_115_CURB_21_01(13,:))/5;

%% Read in the data from the CarSim FFT 114 Run1214

87 fid = fopen(’FFT114_Run1214.bin’,’r’);

% bin file associated with the Michelin’s raw RWA

% data treated in CARSIM

D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT114_Run1214=zeros(10,110); for m=1:110 % 104 corresponds to the number of samples

% per channel

FFT114_Run1214(:,m)=D(((10*(m-1))+1):(10*m));

% 6 corresponds to the number of channels in the ERD file end

%% Read in the data from the CarSim FFT 114 Run1349 fid = fopen(’FFT115_Run1349.bin’,’r’);

% bin file associated with the Michelin’s raw RWA

% data treated in CARSIM

D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT115_Run1349=zeros(17,110); for m=1:110 % 110 corresponds to the number of samples

% per channel

FFT115_Run1349(:,m)=D(((17*(m-1))+1):(17*m));

% 17 corresponds to the number of channels in the ERD file end

88 %% Used for report to Michelin and BMW

% figure

% subplot(2,2,1)

% plot(FFT114_Run1214(1,:),FFT114_Run1214(5,:),’-b’,...

% FFT_115_CURB_21_Frequency_Average,FFT_115_CURB_21_YRGain_Average,...

% ’-g’,FFT115_Run1349(1,:),FFT115_Run1349(9,:),’-r’,’linewidth’,1.5)

% ylabel(’Yaw Rate Gain (1/s)’,’fontweight’,’bold’,’fontsize’,12)

% legend(’CarSim @ 2.1 bar Current TF’,’LPG Test @ 2.1 bar’,...

%’CarSim @ 2.1 bar Previous TF’,’location’,’southwest’)

% grid on

% xlim([0.25 4])

% title(’Yaw Rate Gain/Road Wheel Angle (C+D)’,’fontweight’,...

% ’bold’,’fontsize’,12)

% subplot(2,2,3)

% plot(FFT114_Run1214(1,:),FFT114_Run1214(6,:),’-b’,

% FFT_115_CURB_21_Frequency_Average,FFT_115_CURB_21_YRPhase_Average,...

%’-g’,FFT115_Run1349(1,:), FFT115_Run1349(10,:),’-r’,’linewidth’,1.5)

% xlabel(’Frequency (Hz)’,’fontweight’,’bold’,’fontsize’,12)

% ylabel(’Yaw Rate Phase (deg)’,’fontweight’,’bold’,’fontsize’,12)

% legend(’CarSim @ 2.1 bar Current TF’,’LPG Test @ 2.1 bar’,...

%’CarSim @ 2.1 bar Previous TF’,’location’,’southwest’)

% grid

% xlim([0.25 4])

% title(’Yaw Rate Phase/Road Wheel Angle (C+D)’,’fontweight’,...

%’bold’,’fontsize’,12)

89 % subplot(2,2,2)

% plot(FFT114_Run1214(1,:),FFT114_Run1214(8,:),’-b’,...

% FFT_115_CURB_21_Frequency_Average,FFT_115_CURB_21_AyGain_Average,...

% ’-g’,FFT115_Run1349(1,:),FFT115_Run1349(12,:),’-r’,’linewidth’,1.5)

% ylabel(’Lateral Accel. Gain (g/deg)’,’fontweight’,’bold’,...

% ’fontsize’,12)

% legend(’CarSim @ 2.1 bar Current TF’,’LPG Test @ 2.1 bar’,...

% ’CarSim @ 2.1 bar Previous TF’,’location’,’northeast’)

% grid

% xlim([0.25 4])

% title(’Lateral Accel. Gain/Road Wheel Angle (C+D)’,...

% ’fontweight’,’bold’,’fontsize’,12)

% subplot(2,2,4)

% plot(FFT114_Run1214(1,:),FFT114_Run1214(9,:),’-b’,...

% FFT_115_CURB_21_Frequency_Average,FFT_115_CURB_21_AyPhase_Average,...

% ’-g’,FFT115_Run1349(1,:),FFT115_Run1349(13,:),’-r’,’linewidth’,1.5)

% xlabel(’Frequency (Hz)’,’fontweight’,’bold’,’fontsize’,12)

% ylabel(’Lateral Accel. Phase (deg)’,’fontweight’,’bold’,...

% ’fontsize’,12)

% legend(’CarSim @ 2.1 bar Current TF’,’LPG Test @ 2.1 bar’,...

% ’CarSim @ 2.1 bar Previous TF’,’location’,’northwest’)

% grid

% xlim([0.25 4])

% title(’Lateral Accel. Phase/Road Wheel Angle (C+D)’,...

% ’fontweight’,’bold’,’fontsize’,12)

% print -djpeg -r300 FRF_21barC_D

%%%%

90 % data=[-55.9 2.49 7.23 0.49;-60.3 2.00 7.36 0.49;-60.7 1.6 9.6 0.5]’;

% legend1={’CarSim @ 2.1 bar Current TF’,’LPG Test @ 2.1 bar’,...

% ’CarSim @ 2.1 bar Previous TF’};

% axes_label = {’Lateral Accel Phase @ 1 Hz’ ’(deg)’;...

% ’Yaw Rate Bandwidth’ ’(Hz)’;’SS Yaw Rate Gain’ ’(deg/s)/deg’;...

% ’Yaw Rate Damping Ratio’ “};

% [f, o] = spider(data,’LPG Test Data Compared to CarSim ...

% Simulation (C+D)’,

% [-80 -40 ;0 3.5;0 10;0 0.6],axes_label,legend1);

% print -djpeg -r300 spider_21barC_D

%%%%%%%%%%%%%%%%%%%%%%%%

% Used for Thesis validation

figure subplot(2,2,1) plot(FFT114_Run1214(1,:),FFT114_Run1214(5,:),’-.b’,...

FFT_115_CURB_21_Frequency_Average,FFT_115_CURB_21_YRGain_Average,...

’-k’,’linewidth’,1.5) ylabel(’Yaw Rate/RWA, Gain (1/s)’,’fontweight’,’bold’,...

’fontsize’,12) legend(’Simulation’,’Track Test’,’location’,’southwest’) grid on xlim([0.25 4])

% title(’Yaw Rate Gain/Road Wheel Angle’, ’fontweight’, ’bold’,...

’fontsize’,12)

91 subplot(2,2,3) plot(FFT114_Run1214(1,:),FFT114_Run1214(6,:),’-.b’,...

FFT_115_CURB_21_Frequency_Average,FFT_115_CURB_21_YRPhase_Average...

,’-k’,’linewidth’,1.5) % xlabel(’Frequency (Hz)’,’fontweight’,’bold’,’fontsize’,12) ylabel(’Yaw Rate/RWA, Phase (deg)’,’fontweight’,’bold’,...

’fontsize’,12) legend(’Simulation’,’Track Test’,’location’,’southwest’) grid xlim([0.25 4]) % title(’Yaw Rate Phase/Road Wheel Angle’,’fontweight’,’bold’,...

’fontsize’,12)

subplot(2,2,2) plot(FFT114_Run1214(1,:),FFT114_Run1214(8,:),’-.b’,...

FFT_115_CURB_21_Frequency_Average,FFT_115_CURB_21_AyGain_Average...

,’-k’,’linewidth’,1.5)% ylabel(’Lateral Accel./RWA, Gain (g/deg)’,’fontweight’,’bold’,...

’fontsize’,12) legend(’Simulation’,’Track Test’,’location’,’northeast’)% grid xlim([0.25 4]) % title(’Lateral Accel. Gain/Road Wheel Angle’,’fontweight’,...

’bold’,’fontsize’,12)

subplot(2,2,4) plot(FFT114_Run1214(1,:),FFT114_Run1214(9,:),’-.b’,...

FFT_115_CURB_21_Frequency_Average, FFT_115_CURB_21_AyPhase_Average,...

92 ’-k’,’linewidth’,1.5)%

xlabel(’Frequency (Hz)’,’fontweight’,’bold’,’fontsize’,12)

ylabel(’Lateral Accel./RWA, Phase (deg)’,’fontweight’,’bold’,...

’fontsize’,12)

legend(’Simulation’,’Track Test’,’location’,’northwest’)%

grid xlim([0.25 4]) %

title(’Lateral Accel. Phase/Road Wheel Angle’,’fontweight’,...

’bold’,’fontsize’,12)

print -djpeg -r100 FRF_Thesis_Sim_LPG

data=[-55.9 2.49 7.23 0.49;-60.3 2.00 7.36 0.49]’; %

legend1={’Simulation’,’Track Test’};%

axes_label = {’Lateral Accel Phase @ 1 Hz’ ’(deg)’;...

’Yaw Rate Bandwidth’ ’(Hz)’;’SS Yaw Rate Gain’ ’(deg/s)/deg’;...

’Yaw Rate Damping Ratio’ “};

[f, o] = spider(data,“,[-80 -40 ;0 3.5;0 10;0 0.6],axes_label,legend1);

%%%%%%%%%%%%%%%%%%%%%%%%%%

F.2 TWEEL_Lateral_Force.m

%%%%%%%%%%%%%%%%%

% Program used to calculate the lateral force vs.

% slip angle curves for the Nominal Goodyear

% tire and each of the three hypothetical

% TWEELs

%%%%%%%%%%%%%%%%

clear all

close all

93 clc

%Lateral Tire force estimation

%Lateral Force Symbol = Y, Units = N

%Slip Angle Symbol = delta, Units = deg

%Vertical Load Symbol = Z, Units = kN

%Camber Angle Symbol = gamma, Units = deg

%Pajecka coefficients for OE Mini Goodyear Tire

a0=1.2252; a1=-40.0094; a2=1114.4301; a3=1992.8228; a4=10.2723; a5=0.00531; a6=-0.54271; a7=1.0502; a8=-0.072587; a9=-0.015939; a10=-0.19808; a111=15.4667; a112=-3.5313; a12=37.9298; a13=-43.3789;

%Pacjeka Magic Formula for Mini OE tire

Z=0.5:0.5:7; % range of vertical loads

ZN=Z*1000; %vertical loads in N

94 delta=0:0.1:9.1; %slip angle

gamma_OE=0; % camber angle

Y_OE=zeros(length(delta),length(Z)); %preallocate for i=1:length(delta)

for k=1:length(Z)

D_OE=((a1*Z(k))+a2)*Z(k);

BCD_OE=(a3*sin(2*atan(Z(k)/a4)))*(1-a5*abs(gamma_OE));

C_OE=a0;

B_OE=BCD_OE/(C_OE*D_OE);

E_OE=a6*Z(k)+a7;

phi_OE(i)=(1-E_OE)*(delta(i))+E_OE/B_OE*atan(B_OE*(delta(i)));

Y_OE(i,k)=D_OE*sin(C_OE*atan(B_OE*phi_OE(i)));

end end

figure plot(delta,Y_OE,’linewidth’,1.5) ylabel(’Lateral Force, N’,’fontweight’,’bold’) xlabel(’Slip Angle, deg’,’fontweight’,’bold’)

%title(’OE Tire Lateral Force versus Slip Angle’,’fontweight’,’bold’) legend(’0.5 kN’,’1.0 kN’,’1.5 kN’,’2.0 kN’,’2.5 kN’, ’3.0 kN’,...

’3.5 kN’, ’4.0 kN’,’4.5 kN’,’5.0 kN’,’5.5 kN’,’6.0 kN’,’6.5 kN’,...

’7.0 kN’,’Location’,’best’) grid on

95 %% Pacjeka Magic Formula for TWEEL B

gamma_1=0; %camber change versus load

a3_run1=1392.9544; % Pacejka coefficients a4_run1=8.9047; % for the 3 Series TWEEL

x3_1=1.2; %percent changing original Pacejka x4_1=0.76; %coefficients for 3 Series TWEEL

a3_1=a3_run1*x3_1; % Modifiied Pacejka coefficients a4_1=a4_run1*x4_1; % for a new load insensitive TWEEL

Y_1=zeros(length(delta),length(Z)); %preallocate for i=1:length(delta) %

for k=1:length(Z)

D_1=((a1*Z(k))+a2)*Z(k);

BCD_1=(a3_1*sin(2*atan(Z(k)/a4_1)))*(1-a5*abs(gamma_1));

C_1=a0;

B_1=BCD_1/(C_1*D_1);

E_1=a6*Z(k)+a7;

phi_1=(1-E_1)*(delta(i))+E_1/B_1*atan(B_1*(delta(i)));

Y_1(i,k)=D_1*sin(C_1*atan(B_1*phi_1));

end end

96 figure plot(delta,Y_1,’linewidth’,1.5) ylabel(’Lateral Force, N’,’fontweight’,’bold’) xlabel(’Slip Angle, deg’,’fontweight’,’bold’) title(’Load Sensitive TWEEL B Lateral Force versus Slip Angle’,...

’fontweight’,’bold’) legend(’0.5 kN’,’1.0 kN’,’1.5 kN’,’2.0 kN’,’2.5 kN’, ’3.0 kN’,...

’3.5 kN’,’4.0 kN’,’4.5 kN’,’5.0 kN’,’5.5 kN’,’6.0 kN’,’6.5 kN’,...

’7.0 kN’,’Location’,’best’) grid on

%% Pacjeka Magic Formula for TWEEL C

gamma_1=0; %camber change versus load

x3_2=1.15; % percent changing original Pacejka x4_2=0.55; % coefficients for 3 Series TWEEL a3_2=a3_run1*x3_2; % Modifiied Pacejka coefficients a4_2=a4_run1*x4_2; % for a new load insensitive TWEEL

Y_2=zeros(length(delta),length(Z)); %preallocate for i=1:length(delta)

for k=1:length(Z)

D_2=((a1*Z(k))+a2)*Z(k);

BCD_2=(a3_2*sin(2*atan(Z(k)/a4_2)))*(1-a5*abs(gamma_1));

C_2=a0;

B_2=BCD_2/(C_2*D_2);

E_2=a6*Z(k)+a7;

97 phi_2=(1-E_2)*(delta(i))+E_2/B_2*atan(B_2*(delta(i)));

Y_2(i,k)=D_2*sin(C_2*atan(B_2*phi_2));

end end

figure plot(delta,Y_2,’linewidth’,1.5) ylabel(’Lateral Force, N’,’fontweight’,’bold’) xlabel(’Slip Angle, deg’,’fontweight’,’bold’) title(’Load Sensitive TWEEL C Lateral Force versus Slip Angle’,...

’fontweight’,’bold’) legend(’0.5 kN’,’1.0 kN’,’1.5 kN’,’2.0 kN’,’2.5 kN’, ’3.0 kN’,...

’3.5 kN’, ’4.0 kN’,’4.5 kN’,’5.0 kN’,’5.5 kN’,’6.0 kN’,’6.5 kN’,...

’7.0 kN’,’Location’,’best’) grid on

%% Pacjeka Magic Formula for TWEEL A

gamma_1=0; %camber change versus load

x3_3=0.94; % percent changing original Pacejka x4_3=0.6; % coefficients for 3 Series TWEEL

a3_3=a3_run1*x3_3; % Modifiied Pacejka coefficients a4_3=a4_run1*x4_3; % for a new load insensitive TWEEL

Y_3=zeros(length(delta),length(Z)); %preallocate for i=1:length(delta)

98 for k=1:length(Z)

D_3=((a1*Z(k))+a2)*Z(k);

BCD_3=(a3_3*sin(2*atan(Z(k)/a4_3)))*(1-a5*abs(gamma_1));

C_3=a0;

B_3=BCD_3/(C_3*D_3);

E_3=a6*Z(k)+a7;

phi_3=(1-E_3)*(delta(i))+E_3/B_3*atan(B_3*(delta(i)));

Y_3(i,k)=D_3*sin(C_3*atan(B_3*phi_3));

end end

figure plot(delta,Y_3,’linewidth’,1.5) ylabel(’Lateral Force, N’,’fontweight’,’bold’) xlabel(’Slip Angle, deg’,’fontweight’,’bold’) title(’Load Sensitive TWEEL A Lateral Force versus Slip Angle’,...

’fontweight’,’bold’) legend(’0.5 kN’,’1.0 kN’,’1.5 kN’,’2.0 kN’,’2.5 kN’, ’3.0 kN’,...

’3.5 kN’, ’4.0 kN’,’4.5 kN’,’5.0 kN’,’5.5 kN’,’6.0 kN’,’6.5 kN’,...

’7.0 kN’,’Location’,’best’) grid on

%calculate mu Fy/Fz

% calculate for j=1:k

latss1(j)=Y_1(92,j);

99 mu_1(j)=latss1(j)/ZN(j);

latss2(j)=Y_2(92,j);

mu_2(j)=latss2(j)/ZN(j);

latss3(j)=Y_3(92,j);

mu_3(j)=latss3(j)/ZN(j); end

muavg_1=mean(mu_1); muavg_2=mean(mu_2); muavg_3=mean(mu_3);

fprintf(’Mu average for TWEEL #6, Fy/Fz %.3f \n’,muavg_3) disp(’ ’) fprintf(’Mu average for TWEEL #7, Fy/Fz %.3f \n’,muavg_1) disp(’ ’) fprintf(’Mu average for TWEEL #8, Fy/Fz %.3f \n’,muavg_2) disp(’ ’)

%% plot Calpha vs. load for each of the tires

for i=1:length(Z)

BCD_OE(i)=(a3*sin(2*atan(Z(i)/a4))); end

for i=1:length(Z)

BCD_1(i)=(a3_1*sin(2*atan(Z(i)/a4_1))); end

100 for i=1:length(Z)

BCD_2(i)=(a3_2*sin(2*atan(Z(i)/a4_2))); end

for i=1:length(Z)

BCD_3(i)=(a3_3*sin(2*atan(Z(i)/a4_3))); end

p_OE=polyfit(Z,BCD_OE,2); % find the polynomial p_1=polyfit(Z,BCD_1,2); % fits to the Calpha p_2=polyfit(Z,BCD_2,2); % vs. load curves p_3=polyfit(Z,BCD_3,2);

fit_OE=polyval(p_OE,Z); % uses the fit to find the fit_1=polyval(p_1,Z); % Calpha vs. load and fit_2=polyval(p_2,Z); % compare to the actual curve fit_3=polyval(p_3,Z);

figure subplot(2,2,1) plot(Z,BCD_OE,Z,fit_OE) grid on legend(’Original Curve’,’Fit’,’location’,’best’)

subplot(2,2,2) plot(Z,BCD_1,Z,fit_1)

101 grid on legend(’Original Curve’,’Fit’,’location’,’best’)

subplot(2,2,3) plot(Z,BCD_2,Z,fit_2) grid on legend(’Original Curve’,’Fit’,’location’,’best’)

subplot(2,2,4) plot(Z,BCD_3,Z,fit_3) grid on legend(’Original Curve’,’Fit’,’location’,’best’)

% disp(’C2 for OE’)

% disp(p_OE(1))

% disp(’ ’)

% disp(’C2 for TWEEL #3’)

% disp(p_1(1))

% disp(’ ’)

% disp(’C2 for TWEEL #4’)

% disp(p_2(1))

% disp(’ ’)

% percent_1=(1-p_1(1)/p_OE(1))*100;

% percent_2=(1-p_2(1)/p_OE(1))*100;

%mark each load condition on the plot x1=ones(1,100)*2600; %vertical line to show Rear C+D

102 y1=linspace(800,1400); x2=ones(1,100)*3300; %vertical line to show Rear GVW y2=linspace(1000,1600); x3=ones(1,100)*3900; %vertical line to show Front C+D y3=linspace(1000,1700); x4=ones(1,100)*4300; %vertical line to show Front GVW y4=linspace(1100,1700);

% figure

% plot(ZN,BCD_OE,’k-’,ZN,BCD_3,’b-.’,ZN,BCD_1,’g--’,ZN,BCD_2,...

% ’r:’,x1,y1,’-k’,x3,y3,’-k’,’linewidth’,2)

% xlabel(’Vertical Load Fz (N)’,’fontweight’,’bold’,’fontsize’,13)

% ylabel(’Cornering Stiffness (N/deg)’,’fontweight’,...

% ’bold’,’fontsize’,13)

% title(’Cornering Stiffness vs Vertical Load’,...

% ’fontweight’,’bold’,’fontsize’,13)

% grid on

% legend(’OE Mini Tire’,’TWEEL A’,’TWEEL B’,’TWEEL C’,...

% ’location’,’southeast’)

% text(1000,1700,’Front C+D’,’fontweight’,’bold’,’fontsize’,13)

% text(3800,800,’Rear GVW’,’fontweight’,’bold’,’fontsize’,13)

% text(5400,1100,’Front GVW’,’fontweight’,’bold’,’fontsize’,13)

% text(3300,500,’Rear C+D’,’fontweight’,’bold’,’fontsize’,13)

% print -djpeg -r200 Calpha_vs_load_TWEEL_Thesis

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

figure plot(ZN,BCD_OE,’k-’,ZN,BCD_3,’b-.’,ZN,BCD_1,’g--’,ZN,BCD_2,...

103 ’r:’,x1,y1,’-k’,x3,y3,’-k’,’linewidth’,2) xlabel(’Vertical Load Fz (N)’,’fontweight’,’bold’,’fontsize’,13) ylabel(’Cornering Stiffness (N/deg)’,’fontweight’,’bold’,’fontsize’,13)

% title(’Cornering Stiffness vs Vertical Load’,’fontweight’,...

’bold’,’fontsize’,13) grid on legend(’OE Tire’,’Tire 1’,’Tire 2’,’Tire 3’,’location’,’southeast’) text(1000,1700,’Front C+D’,’fontweight’,’bold’,’fontsize’,13)

%text(3800,800,’Rear GVW’,’fontweight’,’bold’,’fontsize’,13)

%text(5400,1100,’Front GVW’,’fontweight’,’bold’,’fontsize’,13) text(3300,500,’Rear C+D’,’fontweight’,’bold’,’fontsize’,13)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

CalphaF=(a3*sin(2*atan(3.9/a4)));

CalphaR=(a3*sin(2*atan(2.6/a4)));

CalphaF1=(a3_1*sin(2*atan(3.9/a4_1)));

CalphaR1=(a3_1*sin(2*atan(2.6/a4_1)));

CalphaF2=(a3_2*sin(2*atan(3.9/a4_2)));

CalphaR2=(a3_2*sin(2*atan(2.6/a4_2)));

CalphaF3=(a3_3*sin(2*atan(3.9/a4_3)));

CalphaR3=(a3_3*sin(2*atan(2.6/a4_3)));

disp(’ ’) fprintf(’Total Front and Rear Cornering Stiffness Nominal

OE Tire, N/deg %. 1f \n’,Calp haF+CalphaR) disp(’ ’) fprintf(’Front/Rear Split of Cornering Stiffness Nominal

OE Tire %.3f \n’,CalphaF/CalphaR)

104 disp(’ ’)

fprintf(’Total Front and Rear Cornering Stiffness

TWEEL B, N/deg %.1f \n’,CalphaF1+CalphaR1)

disp(’ ’)

fprintf(’Front/Rear Split of Cornering Stiffness

TWEEL B %.3f \n’,CalphaF1/CalphaR1)

disp(’ ’)

fprintf(’Total Front and Rear Cornering Stiffness

TWEEL C, N/deg %.1f \n’,CalphaF2+CalphaR2)

disp(’ ’)

fprintf(’Front/Rear Split of Cornering Stiffness

TWEEL C %.3f \n’,CalphaF2/CalphaR2)

disp(’ ’)

fprintf(’Total Front and Rear Cornering Stiffness

TWEEL A, N/deg %.1f \n’,CalphaF3+CalphaR3)

disp(’ ’)

fprintf(’Front/Rear Split of Cornering Stiffness

TWEEL A %.3f \n’,CalphaF3/CalphaR3)

disp(’ ’)

F.3 TWEEL_Aligning_Moment.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Program used for calculating aligning

% moment of BMW 3 Series TWEEL which

% is used as the aligning moment for

% the Mini

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

105 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% When Michelin ran 3 Series TWEEL it was

% run 3 times due to programming of flat

% track machine. The numbers after the

% coefficient signifies which run it was.

% The numbers stand for the pressure

% reading which is how the flat machine

% works. So c0_18 is the first coefficient

% for what the machine thinks is 1.8 bar.

% This has no significance other than to

% show how the aligning moment decreases

% with wear.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all close all clc

%Aligning Torque

%Aligning Torque Symbol = N, Units = N-m

%Slip Angle Symbol = delta, Units = deg

%Vertical Load Symbol = Z, Units = kN

%Camber Angle Symbol = gamma, Units = deg

%Pajecka coefficients for 3 Series TWEEL

c0_18=3.2132; c1_18=2.1634;

106 c2_18=1.2014; c3_18=-0.79116; c4_18=-1.4124; c5_18=0; c6_18=0; c7_18=-0.16576; c8_18=0.86122; c9_18=-1.4411; c10_18=-0.091491; c11_18=0.22728; c12_18=-0.022053; c13_18=0.11096; c14_18=0.42018; c15_18=-2.6859; c16_18=0.39968; c17_18=2.4225;

%% Run 1

Z=0.75:0.5:5.75; % range of vertical loads

ZN=Z*1000; %vertical load in N delta=0:0.1:9.1; %slip angle tranpdelta=delta’;

%Aligning Moment calculations using Pacekja formulas for i=1:length(delta)

for k=1:length(Z)

gamma(k)=0;

107 D=((c1_18*Z(k))+c2_18)*Z(k);

BCD=(c3_18*Z(k)+c4_18)*Z(k)*exp(-c5_18*Z(k))...

*(1-c6_18*abs(gamma(k)));

C=c0_18;

B=BCD/(C*D);

E=(c7_18*Z(k)^2+c8_18*Z(k)+c9_18)*(1-c10_18*abs(gamma(k)));

phi=(1-E)*delta(i)+E/B*atan(B*delta(i));

N_1(i,k)=-D*sin(C*atan(B*phi));

end end

figure plot(delta,N_1,’linewidth’,2) ylabel(’Aligning Moment, N-m’,’fontweight’,’bold’) xlabel(’Slip Angle, deg’,’fontweight’,’bold’)

%title(’TWEEL Run 1 Aligning Moment versus Slip Angle’,...

% ’fontweight’,’bold’) legend(’0.75 kN’,’1.25 kN’,’1.75 kN’,’2.25 kN’,’2.75 kN’,’3.25 kN’,...

’3.75 kN’,’4.25 kN’,’4.75 kN’,’5.25 kN’,’5.75 kN’,’Location’,’best’) grid on

%% Run 2

%Aligning Moment calculations using Pacekja formulas

c0_24=3.3569; c1_24=2.2141; c2_24=0.95712; c3_24=-0.78064;

108 c4_24=-1.3766; c5_24=0; c6_24=0; c7_24=-0.13542; c8_24=0.58656; c9_24=-0.74952; c10_24=-0.092824; c11_24=0.21832; c12_24=-0.019861; c13_24=0.18277; c14_24=0.37042; c15_24=-2.4142; c16_24=0.48819; c17_24=4.963;

for i=1:length(delta)

for k=1:length(Z)

gamma(k)=0;

D=((c1_24*Z(k))+c2_24)*Z(k);

BCD=(c3_24*Z(k)+c4_24)*Z(k)*exp(-c5_24*Z(k))...

*(1-c6_24*abs(gamma(k)));

C=c0_24;

B=BCD/(C*D);

E=(c7_24*Z(k)^2+c8_24*Z(k)+c9_24)*(1-c10_24*abs(gamma(k)));

phi=(1-E)*delta(i)+E/B*atan(B*delta(i));

N_2(i,k)=-D*sin(C*atan(B*phi));

end

109 end figure plot(delta,N_2,’linewidth’,2) ylabel(’Aligning Moment, N-m’,’fontweight’,’bold’) xlabel(’Slip Angle, deg’,’fontweight’,’bold’) title(’TWEEL Run 2 Aligning Moment versus Slip Angle’,...

’fontweight’,’bold’) legend(’0.75 kN’,’1.25 kN’,’1.75 kN’,’2.25 kN’,’2.75 kN’,’3.25 kN’,...

’3.75 kN’,’4.25 kN’,’4.75 kN’,’5.25 kN’,’5.75 kN’,’Location’,’best’) grid on

%% Run 3

%Aligning Moment calculations using Pacekja formulas

c0_30=3.2817; c1_30=2.2078; c2_30=1.2771; c3_30=-0.77012; c4_30=-1.3408; c5_30=0; c6_30=0; c7_30=-0.1052; c8_30=0.30701; c9_30=-0.54349; c10_30=-0.075644; c11_30=0.21944; c12_30=0.00011484; c13_30=0.12673;

110 c14_30=0.37153; c15_30=-2.388; c16_30=0.60807; c17_30=5.9436;

for i=1:length(delta)

for k=1:length(Z)

gamma(k)=0;

D=((c1_30*Z(k))+c2_30)*Z(k);

BCD=(c3_30*Z(k)+c4_30)*Z(k)*exp(-c5_30*Z(k))...

*(1-c6_30*abs(gamma(k)));

C=c0_30;

B=BCD/(C*D);

E=(c7_30*Z(k)^2+c8_30*Z(k)+c9_30)*(1-c10_30*abs(gamma(k)));

phi=(1-E)*delta(i)+E/B*atan(B*delta(i));

N_3(i,k)=-D*sin(C*atan(B*phi));

end end

figure plot(delta,N_3,’linewidth’,2) ylabel(’Aligning Moment, N-m’,’fontweight’,’bold’) xlabel(’Slip Angle, deg’,’fontweight’,’bold’) title(’TWEEL Run 3 Aligning Moment versus Slip Angle’,...

’fontweight’,’bold’) legend(’0.75 kN’,’1.25 kN’,’1.75 kN’,’2.25 kN’,’2.75 kN’,’3.25 kN’,...

’3.75 kN’,’4.25 kN’,’4.75 kN’,’5.25 kN’,’5.75 kN’,’Location’,’best’)

111 grid on

F.4 Spider Plot Programs

Written by Michael Arant of Michelin North America and a Clemson graduate student.

F.4.1 color_index.m

function [val] = color_index(len)

% get unique colors

% function [val] = color_index(len)

% inputs 1

% len number of objects class integer

% outputs 1

% val color vector class real

% michael arant

if nargin < 1 | nargout < 1; help color_index; error(’I / O error’);

end

if len == 1

val = [0 0 0];

else

% initial color opsibilities (no white)

col = [ 0 0 0

0 0 1

112 0 1 1

0 1 0

1 1 0

1 0 1

1 0 0];

switch len

case 1, col([2 3 4 5 6 7],:) = [];

case 2, col([1 2 3 5 6],:) = [];

case 3, col([1 3 5 6],:) = [];

case 4, col([3 5 6],:) = [];

case 5, col([5 6],:) = [];

case 6, col(6,:) = [];

end

% number of requested colors

val = zeros(len,3); val(:,3) = linspace(0,1,len)’;

val(:,1) = interp1q(linspace(0,1,size(col,1))’,col(:,1),val(:,3));

val(:,2) = interp1q(linspace(0,1,size(col,1))’,col(:,2),val(:,3));

val(:,3) = interp1q(linspace(0,1,size(col,1))’,col(:,3),val(:,3));

end

F.4.2 isint.m

function [res] = isint(val)

% determines if value is an integer

% function [res] = isint(val)

113 %

% inputs 1

% val value to be checked class real

% outputs 1

% res result (1 is integer, 0 is not) class integer

% michael arant may 15, 2004

if nargin < 1; help isint; error(’I / O error’); end

% check for real number

if isreal(val) & isnumeric(val)

% check for integer

if round(val) == val

res = 1;

else

res = 0;

end

else

res = 0;

end

F.4.3 spider.m

function [f, ca, o] = spider(data,tle,rng,lbl,leg,f)

% create a spider plot for ranking the data

% function [f, ca, o] = spider(data,tle,rng,lbl,leg,f)

114 % inputs 6 - 5 optional

% data input data (# axes (M) x # data sets (N)) class real

% tle spider plot title class char

% rng peak range of the data (Mx1 or Mx2) class real

% lbl cell vector axes names (Mxq) in [name unit] pairs class cell

% leg data set legend identification (1xN) class cell

% f figure handle or plot handle class real

% outptus 3 - 3 optional

% f figure handle class integer

% x axes handle class real

% o series object handles class real

% michael arant - jan 30, 2008

% data check if nargin < 1; help spider; error(’Need data to plot’); end

% size segments and number of cases

[r c] = size(data);

% exit for too few axes if r < 3

errordlg(’Must have at least three measuremnt axes’)

error(’Program Termination: Must have a minimum of three axes’) end

% title if ~exist(’tle’,’var’) || isempty(tle) || ~ischar(tle)

115 tle = “; end

% check for maximum range if ~exist(’rng’,’var’) || isempty(rng) || ~isreal(rng)

% no range given or range is in improper format

% define new range

rng = [min([min(data,[],2) zeros(r,1)],[],2) max(data,[],2)];

% check for negative minimum values

if ~isempty(ismember(-1,sign(data)))

% negative value found - adjust minimum range

for ii = 1:r

% negative range for axis ii - set new minimum

if min(data(ii,:)) < 0

rng(ii,1) = min(data(ii,:)) - ...

0.25 * (max(data(ii,:)) - min(data(ii,:)));

end

end end

elseif size(rng,1) ~= r

% insuffent range definition

uiwait(msgbox(char(’Range size must be Mx1 -

number of axes x 1’, ...

sprintf(’%g axis ranges defined, %g axes exist’,

size(rng,1),r))))

116 error(sprintf(’%g axis ranges defined, %g axes exist’,

size(rng,1),r)) elseif size(rng,2) == 1

% assume range is a maximum range - define minimum

rng = sort([min([zeros(r,1) min(data,[],2) - ...

0.25 * (max(data,[],2) - min(data,[],2))],[],2) rng],2);

% elseif size(rng,1) ~= r || size(rng,2) ~= 2

% % invalid range

% uiwait(msgbox(’Range must have same number of rows

% as data set’))

% error(’Invalid range specification’) end

% check for axis labels if ~exist(’lbl’,’var’) || isempty(lbl)

% no labels given - define a default lable

lbl = cell(r,1);

for ii = 1:r;

lbl(ii) = cellstr(sprintf(’Axis %g’,ii));

end elseif size(lbl,1) ~= r

if size(lbl,2) == r

lbl = lbl’;

else

117 uiwait(msgbox(char(’Axis labels must be Mx1 -

number of axes x 1’, ...

sprintf(’%g axis labels defined, %g axes exist’,

size(lbl,1),r))))

error(sprintf(’%g axis labels defined,

%g axes exist’,size(lbl,1),r))

end elseif ischar(lbl)

% check for charater labels

lbl = cellstr(lbl); end

if ~exist(’leg’,’var’) || isempty(leg)

% no data legend - define default legend

leg = cell(1,c);

for ii = 1:c;

leg(ii) = cellstr(sprintf(’Set %g’,ii));

end elseif numel(leg) ~= c

uiwait(msgbox(char(’Data set label must be 1XN -

1 x number of sets’, ...

sprintf(’%g data sets labeled, %g exist’,numel(leg),c))))

error(sprintf(’%g data sets labeled, %g exist’,numel(leg),c))

118 end

% check for figure or axes if ~exist(’f’,’var’)

% no figure or axes requested - generate new ones

f = figure; ca = gca(f); cla(ca); hold on elseif ismember(f,get(0,’children’)’)

% existing figure - clear and set up

clf(f); ca = gca(f); cla(ca); hold on elseif isint(f)

% generating a new figure

figure(f); ca = gca(f); cla(ca); hold on else

% may be an axes - may be garbage

try

%is this an axes?

if ismember(get(f,’parent’),get(0,’children’)’)

% existing figure axes - use

ca = f; f = get(f,’parent’); cla(ca); hold on

end

catch

% make new figure and axes

disp(sprintf(’Invalid axes handle %g passed.

Generating new figure’,f))

f = figure; ca = gca(f); cla(ca); hold on

end

119 end

% clear figure and set limits set(ca,’visible’,’off’); set(f,’color’,’w’) set(ca,’xlim’,[-1.25 1.25],’ylim’,[-1.25 1.25]);

axis(ca,’equal’,’manual’) % title text(0,1.25,tle,’horizontalalignment’,’center’,’fontweight’,’bold’);

% define data case colors col = color_index(c);

% scale by range angw = linspace(0,2*pi,r+1)’; mag = (data - rng(:,1) * ones(1,c)) ./ (diff(rng,[],2) * ones(1,c));

% wrap data (close the last axis to the first) ang = angw(1:end-1); angwv = angw * ones(1,c); magw = [mag; mag(1,:)];

% make the plot

% define the axis locations start = [zeros(1,r); cos(ang’)]; stop = [zeros(1,r); sin(ang’)];

% plot the axes plot(ca,start,stop,’color’,’k’,’linestyle’,’-’); axis equal

% plot axes markers inc = 0.25:.25:1; mk = .025 * ones(1,4); tx = 4 * mk; tl = 0:.25:1;

120 % loop each axis ang plot the line markers and labels for ii = 1:r

% plot tick marks

plot(ca,[[cos(ang(ii)) * inc + sin(ang(ii)) * mk]; ...

[cos(ang(ii)) * inc - sin(ang(ii)) * mk]], ...

[[sin(ang(ii)) * inc - cos(ang(ii)) * mk] ;

[sin(ang(ii)) * inc + cos(ang(ii)) * mk]],’color’,’k’)

% label the tick marks

for jj = 1:4

temp = text([cos(ang(ii)) * inc(jj) +% 90 corresponds to the number of samples per channel

% 6 corresponds to the number of channels in the ERD file FFT_126_CURB_21_05(:,m)=D(((6*(m-1))+1):(6*m)); sin(ang(ii)) *

tx(jj)], ...

[sin(ang(ii)) * inc(jj) - cos(ang(ii)) * tx(jj)], ...

num2str(chop(rng(ii,1) + inc(jj)*diff(rng(ii,:)),2)));

% temp = text([cos(ang(ii)) * tl(jj) + sin(ang(ii)) * tx(jj)], ...

% [sin(ang(ii)) * tl(jj) - cos(ang(ii)) * tx(jj)], ...

% num2str(chop(rng(ii,1) + tl(jj)*diff(rng(ii,:)))));

set(temp,’fontsize’,8)

% flip the text alignment for lower axes

if ang(ii) >= pi

set(temp,’HorizontalAlignment’,’right’)

end end

% label each axis

121 temp = text([cos(ang(ii)) * 1.1 + sin(ang(ii)) * 0], ...

[sin(ang(ii)) * 1.1 - cos(ang(ii)) * 0],char(lbl(ii,:)));

% flip the text alignment for right side axes

if ang(ii) > pi/2 && ang(ii) < 3*pi/2

set(temp,’HorizontalAlignment’,’right’)

end

% plot the data

o = polar(ca,angw*ones(1,c),magw);

% apply the legend

temp = legend(ca,o,leg,’location’,’northwest’);

% set the legend location out of the plot window

set(temp,’position’,get(temp,’position’) .* [1.25 1 1 1])

% set the color for each data set

for ii = 1:c

% set(o(ii),’color’,col(ii,:),’linewidth’,2.2)

set(o(1),’color’,’k’,’linestyle’,’-.’,’linewidth’,2.2)

set(o(2),’color’,’b’,’linestyle’,’-’,’linewidth’,2.2)

set(o(3),’color’,’g’,’linestyle’,’--’,’linewidth’,2.2)

set(o(4),’color’,’r’,’linestyle’,’:’,’linewidth’,2.2)

end

F.5 Frequency_Response_Functions.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

122 % Program that takes frequency data from CarSim

% (yaw rate, lateral acceleration, and RWA/SWA

% and converts RWA to SWA and then plots the

% transfer functions and the spider plots

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all close all clc

% read-in order

% transfer function for going from RWA to SWA

% run1=60/40, OE Tire

% run2=60/40, Tire 1

% run3=60/40, Tire 2

% run4=60/40, Tire 3

% run5=55/45, OE Tire

% run6=50/50, OE Tire

% run7=50/50, Tire 1

% run8=50/50, Tire 2

% run9=50/50, Tire 3

% run10=55/45, Tire 1

% run11=55/45, Tire 2

% run12=55/45, Tire 3

%% Read in the data from the CarSim FFT 126 CURB_21_01 fid = fopen(’FFT126_CURB_21_1_20_SWA.bin’,’r’);

% bin file associated with the Michelin’s raw RWA/SWA data

123 % treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

for m=1:90

% 90 corresponds to the number of samples per channel

% 6 corresponds to the number of channels in the ERD file

FFT_126_CURB_21_01(:,m)=D(((6*(m-1))+1):(6*m)); end

%% Read in the data from the CarSim FFT 126 CURB_21_02 fid = fopen(’FFT126_CURB_21_2_20_SWA.bin’,’r’);

% bin file associated with the Michelin’s raw RWA/SWA data

% treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

for m=1:90

% 90 corresponds to the number of samples per channel

% 6 corresponds to the number of channels in the ERD file

FFT_126_CURB_21_02(:,m)=D(((6*(m-1))+1):(6*m)); end

124 %% Read in the data from the CarSim FFT 126 CURB_21_03 fid = fopen(’FFT126_CURB_21_3_20_SWA.bin’,’r’);

% bin file associated with the Michelin’s raw RWA/SWA data

% treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

for m=1:90

% 90 corresponds to the number of samples per channel

% 6 corresponds to the number of channels in the ERD file

FFT_126_CURB_21_03(:,m)=D(((6*(m-1))+1):(6*m)); end

%% Read in the data from the CarSim FFT 126 CURB_21_04 fid = fopen(’FFT126_CURB_21_4_20_SWA.bin’,’r’);

% bin file associated with the Michelin’s raw RWA/SWA data

% treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

for m=1:90

% 90 corresponds to the number of samples per channel

% 6 corresponds to the number of channels in the ERD file

FFT_126_CURB_21_04(:,m)=D(((6*(m-1))+1):(6*m));

125 end

%% Read in the data from the CarSim FFT 126 CURB_21_05 fid = fopen(’FFT126_CURB_21_5_20_SWA.bin’,’r’);

% bin file associated with the Michelin’s raw RWA/SWA

% data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

for m=1:90

% 90 corresponds to the number of samples per channel

% 6 corresponds to the number of channels in the ERD file

FFT_126_CURB_21_05(:,m)=D(((6*(m-1))+1):(6*m)); end

%% Read in the data from the CarSim FFT 133 Run1353 Nominal

% OE Tire and nominal weight distribution fid = fopen(’FFT133_Run1353.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data

% treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

FFT133_Run1353=zeros(10,110);

126 for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1353(:,m)=D(((10*(m-1))+1):(10*m)); end run1=FFT133_Run1353;

%% Read in the data from the CarSim FFT 133 Run1362

% TWEEL A, nominal weight distribution fid = fopen(’FFT133_Run1396.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data

% treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

FFT133_Run1396=zeros(10,110); for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1396(:,m)=D(((10*(m-1))+1):(10*m)); end run2=FFT133_Run1396;

127 %% Read in the data from the CarSim FFT 133 Run1363

% TWEEL B, nominal weight distribution fid = fopen(’FFT133_Run1363.bin’,’r’);

% bin file associated with the Michelin’s raw RWA data

% treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

FFT133_Run1363=zeros(10,110); for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1363(:,m)=D(((10*(m-1))+1):(10*m)); end run3=FFT133_Run1363;

%% Read in the data from the CarSim FFT 133 Run1364

% TWEEL C, nominal weight distribution fid = fopen(’FFT133_Run1364.bin’,’r’);

% bin file associated with the Michelin’s raw RWA

% data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default CARSIM

% stores binary data in this format.

FFT133_Run1364=zeros(10,110);

128 for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1364(:,m)=D(((10*(m-1))+1):(10*m)); end run4=FFT133_Run1364;

%% Read in the data from the CarSim FFT 133

% Run 1400 55/45 weight dis. fid = fopen(’FFT133_Run1403.bin’,’r’);

% bin file associated with the Michelin’s raw RWA

% data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT133_Run1403=zeros(10,110);

for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1403(:,m)=D(((10*(m-1))+1):(10*m)); end run5=FFT133_Run1403;

129 %% Read in the data from the CarSim FFT 133

% Run 1400 50/50 weight dis. fid = fopen(’FFT133_Run1399.bin’,’r’);

% bin file associated with the Michelin’s raw RWA

% data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT133_Run1399=zeros(10,110);

for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1399(:,m)=D(((10*(m-1))+1):(10*m)); end run6=FFT133_Run1399;

%% Read in the data from the CarSim FFT 133

% Run 1404 50/50 weight dis. with tire 1 fid = fopen(’FFT133_Run1404.bin’,’r’);

% bin file associated with the Michelin’s raw

% RWA data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by

% default CARSIM stores binary data in this format.

FFT133_Run1404=zeros(10,110);

130 for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1404(:,m)=D(((10*(m-1))+1):(10*m)); end run7=FFT133_Run1404;

%% Read in the data from the CarSim FFT 133

% Run 1405 50/50 weight dis. with tire 2 fid = fopen(’FFT133_Run1405.bin’,’r’);

% bin file associated with the Michelin’s raw

% RWA data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by

% default CARSIM stores binary data in this format.

FFT133_Run1405=zeros(10,110); for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1405(:,m)=D(((10*(m-1))+1):(10*m)); end run8=FFT133_Run1405;

131 %% Read in the data from the CarSim FFT 133

% Run1406 50/50 weight dis. with tire 3 fid = fopen(’FFT133_Run1406.bin’,’r’);

% bin file associated with the Michelin’s raw

% RWA data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT133_Run1406=zeros(10,110); for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1406(:,m)=D(((10*(m-1))+1):(10*m)); end run9=FFT133_Run1406;

%% Read in the data from the CarSim FFT 133

% Run1407 55/45 weight dis. with tire 1 fid = fopen(’FFT133_Run1407.bin’,’r’);

% bin file associated with the Michelin’s raw

% RWA data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT133_Run1407=zeros(10,110); for m=1:110

132 % 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1407(:,m)=D(((10*(m-1))+1):(10*m)); end run10=FFT133_Run1407;

%% Read in the data from the CarSim FFT 133

% Run1408 55/45 weight dis. with tire 2 fid = fopen(’FFT133_Run1408.bin’,’r’);

% bin file associated with the Michelin’s raw RWA

% data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT133_Run1408=zeros(10,110); for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1408(:,m)=D(((10*(m-1))+1):(10*m)); end run11=FFT133_Run1408;

%% Read in the data from the CarSim FFT 133

133 % Run1409 55/45 weight dis. with tire 3 fid = fopen(’FFT133_Run1409.bin’,’r’);

% bin file associated with the Michelin’s raw

% RWA data treated in CARSIM D=fread(fid,’float32’);

% float32 has been used to decode, since by default

% CARSIM stores binary data in this format.

FFT133_Run1409=zeros(10,110); for m=1:110

% 110 corresponds to the number of samples per channel

% 10 corresponds to the number of channels in the ERD file

FFT133_Run1409(:,m)=D(((10*(m-1))+1):(10*m)); end run12=FFT133_Run1409;

%% Average the Frequency, Gain and Phase for the RWA/SWA

% data from CarSim for the Michelin data

CURB_21_RWA_SWA_Frequency=(FFT_126_CURB_21_05(1,:)+...

FFT_126_CURB_21_04(1,:)+FFT_126_CURB_21_03(1,:)+...

FFT_126_CURB_21_02(1,:)+FFT_126_CURB_21_01(1,:))/5;

CURB_21_RWA_SWA_Gain=(FFT_126_CURB_21_05(4,:)+...

FFT_126_CURB_21_04(4,:)+FFT_126_CURB_21_03(4,:)+...

FFT_126_CURB_21_02(4,:)+FFT_126_CURB_21_01(4,:))/5;

134 CURB_21_RWA_SWA_Phase=(FFT_126_CURB_21_05(5,:)+...

FFT_126_CURB_21_04(5,:)+FFT_126_CURB_21_03(5,:)+...

FFT_126_CURB_21_02(5,:)+FFT_126_CURB_21_01(5,:))/5;

GP=[CURB_21_RWA_SWA_Gain;CURB_21_RWA_SWA_Phase]; freq_rad=(CURB_21_RWA_SWA_Frequency)*(2*pi); load prep.mat prepf = idfilt(prep,[1.57,22]);

% band pass filterd frq in rad/s since input

% freq in prep is in rad/s

spad = etfe(prepf,12,10000);

[mag,phase,w] = bode(spad); mag=squeeze(mag); phase=squeeze(phase); freq_Hz=w/(2*pi); mag1 = interp1(freq_Hz,mag,run1(1,:),’linear’,’extrap’); phase1= interp1(freq_Hz,phase,run1(1,:),’linear’,’extrap’);

% Adjust the max value of j to give you the

% max no. of runs you are analyzing for j=1:12

if j==1

temp=run1;

elseif j==2

temp=run2;

135 elseif j==3

temp=run3;

elseif j==4

temp=run4;

elseif j==5

temp=run5;

elseif j==6

temp=run6;

elseif j==7

temp=run7;

elseif j==8

temp=run8;

elseif j==9

temp=run9;

elseif j==10

temp=run10;

elseif j==11

temp=run11;

elseif j==12

temp=run12;

end

FFT_freq_SWA(j,:)=(temp(1,:));

FFT_Yaw_rateG_SWA(j,:)=mag1.*(temp(5,:));

FFT_Yaw_rateP_SWA(j,:)=phase1+(temp(6,:));

136 % FFT_Yaw_rateG_SWA1(j,:)=CURB_21_RWA_SWA_Gain.*(temp(5,:));

FFT_AyG_SWA(j,:)=mag1.*(temp(8,:));

FFT_AyP_SWA(j,:)=phase1+(temp(9,:));

% FFT_AyP_SWA1(j,:)=CURB_21_RWA_SWA_Phase+(temp(9,:));

end

%% plot the results figure subplot(2,2,1) plot(FFT_freq_SWA(1,:),FFT_Yaw_rateG_SWA(1,:),’k-’,...

FFT_freq_SWA(2,:),FFT_Yaw_rateG_SWA(2,:),’b-.’,...

FFT_freq_SWA(3,:),FFT_Yaw_rateG_SWA(3,:),’g--’,...

FFT_freq_SWA(4,:),FFT_Yaw_rateG_SWA(4,:),’r:’,’linewidth’,1.5) ylabel(’Yaw Rate/SWA, Gain (1/s)’,’fontweight’,’bold’,’fontsize’,12) legend(’Nominal OE Tire’,’TWEEL A’,’TWEEL B’,’TWEEL C’,...

’location’,’southwest’)

grid on

xlim([0.25 4]) title(’Yaw Rate Gain/Road Wheel Angle (C+D)’,...

’fontweight’,’bold’,’fontsize’,12)

subplot(2,2,3) plot(FFT_freq_SWA(1,:),FFT_Yaw_rateP_SWA(1,:),’-k’,...

FFT_freq_SWA(2,:),FFT_Yaw_rateP_SWA(2,:),’-.b’,...

FFT_freq_SWA(3,:),FFT_Yaw_rateP_SWA(3,:),’--g’,...

137 FFT_freq_SWA(4,:),FFT_Yaw_rateP_SWA(4,:),’:r’,’linewidth’,1.5) xlabel(’Frequency (Hz)’,’fontweight’,’bold’,’fontsize’,12) ylabel(’Yaw Rate/SWA, Phase (deg)’,’fontweight’,’bold’,’fontsize’,12) legend(’Nominal OE Tire’,’TWEEL A’,’TWEEL B’,’TWEEL C’,...

’location’,’northeast’) grid xlim([0.25 4]) title(’Yaw Rate Phase/Road Wheel Angle (C+D)’,’fontweight’,...

’bold’,’fontsize’,12)

subplot(2,2,2) plot(FFT_freq_SWA(1,:),FFT_AyG_SWA(1,:),’-k’,FFT_freq_SWA(2,:),...

FFT_AyG_SWA(2,:),’-.b’,FFT_freq_SWA(3,:),FFT_AyG_SWA(3,:),...

’--g’,FFT_freq_SWA(4,:),FFT_AyG_SWA(4,:),’:r’,’linewidth’,1.5) ylabel(’Lateral Accel./SWA, Gain (g/deg)’,’fontweight’,...

’bold’,’fontsize’,12) legend(’Nominal OE Tire’,’TWEEL A’,’TWEEL B’,...

’TWEEL C’,’location’,’northeast’) grid xlim([0.25 4]) title(’Lateral Accel. Gain/Road Wheel Angle (C+D)’,...

’fontweight’,’bold’,’fontsize’,12)

138 subplot(2,2,4) plot(FFT_freq_SWA(1,:),FFT_AyP_SWA(1,:),’-k’,FFT_freq_SWA(2,:),...

FFT_AyP_SWA(2,:),’-.b’,FFT_freq_SWA(3,:),FFT_AyP_SWA(3,:),...

’--g’,FFT_freq_SWA(4,:),FFT_AyP_SWA(4,:),’:r’,’linewidth’,1.5) xlabel(’Frequency (Hz)’,’fontweight’,’bold“bold’,’fontsize’,12) legend(’Nominal OE Tire’,’TWEEL A’,’TWEEL B’,...

’TWEEL C’,’location’,’northeast’) ,’fontsize’,12) ylabel(’Lateral Accel./SWA, Phase (deg)’,’fontweight’,...

’bold’,’fontsize’,12) legend(’Nominal OE Tire’,’TWEEL A’,’TWEEL B’,’TWEEL C’,...

’location’,’southeast’) grid xlim([0.25 4]) title(’Lateral Accel. Phase/Road Wheel Angle (C+D)’,...

’fontweight’,’bold’,’fontsize’,12) print -djpeg -r300 FRF_Thesis_C_D

%% 4 parameters for thesis for k=1:12

SS_yaw_rate_gain(k) = interp1(FFT_freq_SWA(k,:),...

FFT_Yaw_rateG_SWA(k,:),0.28,’linear’,’extrap’); damping_coeff(k)=1/(2*max(FFT_Yaw_rateG_SWA(k,:))/...

SS_yaw_rate_gain(k));

139 yaw_rate_3db_gain(k)=0.7079.*(SS_yaw_rate_gain(k)); tempo=FFT_Yaw_rateG_SWA(k,:); k; s=1; clear high1 clear low1 while s<=max(size(FFT_Yaw_rateG_SWA))

if tempo(s)<(0.7079*SS_yaw_rate_gain(k))

low1(s)=s;

elseif tempo(s)>(0.7079*SS_yaw_rate_gain(k))

high1(s)=s;

else

end s=s+1; end

for w=1:length(low1)

if low1(1)==0;

low1(1)=[];

end end

high1(end); three_db_bandwidth(k)=FFT_freq_SWA(k,low1(1));

Ay_phase_1Hz(k)=FFT_AyP_SWA(k,46);

140 % 46 is the element no. corresponding to 1 Hz

% in frequency vector FFT_freq_SWA end

% data for the changing cornering stiffness

% (OE Tire, Tire 1, Tire 2, Tire3) data=[Ay_phase_1Hz(1) three_db_bandwidth(1) SS_yaw_rate_gain(1)...

damping_coeff(1); Ay_phase_1Hz(2) three_db_bandwidth(2)...

SS_yaw_rate_gain(2) damping_coeff(2); Ay_phase_1Hz(3)...

three_db_bandwidth(3) SS_yaw_rate_gain(3) damping_coeff(3);...

Ay_phase_1Hz(4) three_db_bandwidth(4) SS_yaw_rate_gain(4)...

damping_coeff(4)]’;

data1=[Ay_phase_1Hz(1) three_db_bandwidth(1) SS_yaw_rate_gain(1)...

damping_coeff(1); Ay_phase_1Hz(5) three_db_bandwidth(5)...

SS_yaw_rate_gain(5) damping_coeff(5); Ay_phase_1Hz(6)...

three_db_bandwidth(6) SS_yaw_rate_gain(6) damping_coeff(6)]’;

data12=[Ay_phase_1Hz(1) three_db_bandwidth(1) SS_yaw_rate_gain(1)...

damping_coeff(1); Ay_phase_1Hz(2) three_db_bandwidth(2)...

SS_yaw_rate_gain(2) damping_coeff(2); Ay_phase_1Hz(3)...

three_db_bandwidth(3) SS_yaw_rate_gain(3) damping_coeff(3);...

Ay_phase_1Hz(4) three_db_bandwidth(4) SS_yaw_rate_gain(4)...

damping_coeff(4); Ay_phase_1Hz(5) three_db_bandwidth(5)...

141 SS_yaw_rate_gain(5) damping_coeff(5); Ay_phase_1Hz(6)...

three_db_bandwidth(6) SS_yaw_rate_gain(6) damping_coeff(6);...

Ay_phase_1Hz(7) three_db_bandwidth(7) SS_yaw_rate_gain(7)...

damping_coeff(7); Ay_phase_1Hz(8) three_db_bandwidth(8)...

SS_yaw_rate_gain(8) damping_coeff(8); Ay_phase_1Hz(9)...

three_db_bandwidth(9) SS_yaw_rate_gain(9) damping_coeff(9);...

Ay_phase_1Hz(10) three_db_bandwidth(10) SS_yaw_rate_gain(10)...

damping_coeff(10); Ay_phase_1Hz(11) three_db_bandwidth(11)...

SS_yaw_rate_gain(11) damping_coeff(11); Ay_phase_1Hz(12)...

three_db_bandwidth(12) SS_yaw_rate_gain(12) damping_coeff(12)]’;

legend1={’Nominal OE Tire’,’TWEEL A’,’TWEEL B’,’TWEEL C’}; axes_label = {’Lateral Accel Phase @ 1 Hz’ ’(deg)’;...

’Yaw Rate Bandwidth’ ’(Hz)’; ’SS Yaw Rate Gain’ ’(deg/s)/deg’;...

’Yaw Rate Damping Ratio’ “};

[f, o] = spider(data,“,[-60 -30 ;0 3.5;0.1 0.5;0 0.6],axes_label,...

legend1); print -djpeg -r300 spider_Thesis_TWEELS

%% spider plot for example data1=[Ay_phase_1Hz(1) three_db_bandwidth(1) SS_yaw_rate_gain(1)...

damping_coeff(1)]’; axes_label = {’Lateral Accel Phase @ 1 Hz’ ’(deg)’;...

’Yaw Rate Bandwidth’ ’(Hz)’; ’SS Yaw Rate Gain’ ’(deg/s)/deg’;...

142 ’Yaw Rate Damping Ratio’ “};

[f, o] = spider(data1,’Sensitivity to Load for TWEELs (C+D)’,...

[-70 -50 ;0 3.5;0.1 0.5;0 0.6],axes_label); print -djpeg -r300 spider_Thesis_example

%% plot frequency response functions for weight distrubutions figure subplot(2,2,1) plot(FFT_freq_SWA(1,:),FFT_Yaw_rateG_SWA(1,:),FFT_freq_SWA(5,:),...

FFT_Yaw_rateG_SWA(5,:),’:’,FFT_freq_SWA(6,:),...

FFT_Yaw_rateG_SWA(6,:),’-.’,’linewidth’,1.5) ylabel(’Yaw Rate/SWA, Gain (1/s)’,’fontweight’,’bold’,’fontsize’,12) grid on

% legend(’OE Tire’,’55/45 Dis.’,’50/50 Dis.’,’location’,’northeast’) xlim([0.25 4])

subplot(2,2,3) plot(FFT_freq_SWA(1,:),FFT_Yaw_rateP_SWA(1,:),FFT_freq_SWA(5,:),...

FFT_Yaw_rateP_SWA(5,:),’:’,FFT_freq_SWA(6,:),...

FFT_Yaw_rateP_SWA(6,:),’-.’,’linewidth’,1.5) xlabel(’Frequency (Hz)’,’fontweight’,’bold’,’fontsize’,12) ylabel(’Yaw Rate/SWA, Phase (deg)’,’fontweight’,’bold’,...

’fontsize’,12) grid

% legend(’OE Tire’,’55/45 Dis.’,’50/50 Dis.’,’location’,’northeast’)

143 xlim([0.25 4])

subplot(2,2,2) plot(FFT_freq_SWA(1,:),FFT_AyG_SWA(1,:),FFT_freq_SWA(5,:),...

FFT_AyG_SWA(5,:),’:’,FFT_freq_SWA(6,:),FFT_AyG_SWA(6,:),...

’-.’,’linewidth’,1.5) ylabel(’Lateral Accel./SWA, Gain (g/deg)’,’fontweight’,’bold’,...

’fontsize’,12) grid

% legend(’OE Tire’,’55/45 Dis.’,’50/50 Dis.’,’location’,’northeast’) xlim([0.25 4])

subplot(2,2,4) plot(FFT_freq_SWA(1,:),FFT_AyP_SWA(1,:),FFT_freq_SWA(5,:),...

FFT_AyP_SWA(5,:),’:’,FFT_freq_SWA(6,:),FFT_AyP_SWA(6,:),...

’-.’,’linewidth’,1.5) xlabel(’Frequency (Hz)’,’fontweight’,’bold’,’fontsize’,12) ylabel(’Lateral Accel./SWA, Phase (deg)’,’fontweight’,...

’bold’,’fontsize’,12) grid legend(’OE Tire’,’55/45 Dis.’,’50/50 Dis.’,’location’,’southeast’) xlim([0.25 4])

% % print -djpeg -r300 FRF_

144 %% plot correlation plots load ’C:\Documents and Settings\rpowell\My Documents\Thesis

\IJVD Paper\Journal Article\Normalized_Ji’

J1=[OE_Tire_60_40(1) Tire_1_60_40(1) Tire_2_60_40(1)...

Tire_3_60_40(1) OE_Tire_55_45(1) OE_Tire_50_50(1)...

Tire_1_50_50(1) Tire_2_50_50(1) Tire_3_50_50(1)...

Tire_1_55_45(1) Tire_2_55_45(1) Tire_3_55_45(1)];

J2=[OE_Tire_60_40(2) Tire_1_60_40(2) Tire_2_60_40(2)...

Tire_3_60_40(2) OE_Tire_55_45(2) OE_Tire_50_50(2)...

Tire_1_50_50(2) Tire_2_50_50(2) Tire_3_50_50(2)...

Tire_1_55_45(2) Tire_2_55_45(2) Tire_3_55_45(2)];

J3=[OE_Tire_60_40(3) Tire_1_60_40(3) Tire_2_60_40(3)...

Tire_3_60_40(3) OE_Tire_55_45(3) OE_Tire_50_50(3)...

Tire_1_50_50(3) Tire_2_50_50(3) Tire_3_50_50(3)...

Tire_1_55_45(3) Tire_2_55_45(3) Tire_3_55_45(3)];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure subplot(2,2,4) plot(Ay_phase_1Hz(1),J3(1),’^k’,Ay_phase_1Hz(2),J3(2),’vk’,...

Ay_phase_1Hz(3),J3(3),’k’,...

Ay_phase_1Hz(5),J3(5),’hb’,Ay_phase_1Hz(10),J3(10),’pb’,...

Ay_phase_1Hz(11),J3(11),’xb’,Ay_phase_1Hz(12),J3(12),’+b’,...

Ay_phase_1Hz(6),J3(6),’or’,Ay_phase_1Hz(7),J3(7),’sr’,...

Ay_phase_1Hz(8),J3(8),’.r’,Ay_phase_1Hz(9),J3(9),’*r’,...

145 ’MarkerSize’,10) legend(’60/40, OE Tire’,’60/40, Tire 1’,’60/40, Tire 2’,...

’60/40, Tire 3’,’55/45, OE Tire’,’55/45, Tire 1’,...

’55/45, Tire 2’,’55/45, Tire 3’,’50/50, OE Tire’,...

’50/50, Tire 1’,’50/50, Tire 2’,’50/50, Tire 3’,’location’,...

’best’) xlabel(’Lateral Acceleration Phase Lag at 1 Hz, deg’,...

’fontweight’,’bold’) ylabel(’J3 Normalized’,’fontweight’,’bold’) grid

subplot(2,2,3) plot(three_db_bandwidth(1),J3(1),’^k’,three_db_bandwidth(2),...

J3(2),’vk’,three_db_bandwidth(3),J3(3),’

three_db_bandwidth(4),J3(4),’>k’,three_db_bandwidth(5),...

J3(5),’hb’,three_db_bandwidth(6),J3(6),’or’,...

three_db_bandwidth(7),J3(7),’sr’,three_db_bandwidth(8),...

J3(8),’.r’,three_db_bandwidth(9),J3(9),’*r’,...

three_db_bandwidth(10),J3(10),’pb’,three_db_bandwidth(11),...

J3(11),’xb’,three_db_bandwidth(12),J3(12),’+b’,...

’MarkerSize’,10) xlabel(’Yaw Rate Bandwidth, Hz’,’fontweight’,’bold’) ylabel(’J3 Normalized’,’fontweight’,’bold’) grid

% subplot(3,2,1)

146 % plot(SS_yaw_rate_gain(1),J1(1),’^k’,SS_yaw_rate_gain(2),J1(2),...

% ’vk’,SS_yaw_rate_gain(3),J1(3),’

% J1(4),’>k’,SS_yaw_rate_gain(5),J1(5),’hb’,SS_yaw_rate_gain(6)...

% ,J1(6),’or’,SS_yaw_rate_gain(7),J1(7),’sr’,SS_yaw_rate_gain(8),...

% J1(8),’.r’,SS_yaw_rate_gain(9),J1(9),’*r’,SS_yaw_rate_gain(10),...

% J1(10),’pb’,SS_yaw_rate_gain(11),J1(11),’xb’,...

% SS_yaw_rate_gain(12),J1(12),’+b’,’MarkerSize’,10)

% xlabel(’Steady State Yaw Rate Gain, 1/s’,’fontweight’,’bold’)

% ylabel(’J1 Normalized’,’fontweight’,’bold’)

% grid

% subplot(2,2,1)

% plot(three_db_bandwidth(1),J2(1),’^k’,three_db_bandwidth(2),...

% J2(2),’vk’,three_db_bandwidth(3),J2(3),’

% three_db_bandwidth(4),J2(4),’>k’,three_db_bandwidth(5),J2(5),...

% ’hb’,three_db_bandwidth(6),J2(6),’or’,three_db_bandwidth(7),...

% J2(7),’sr’,three_db_bandwidth(8),J2(8),’.r’,...

% three_db_bandwidth(9),J2(9),’*r’,three_db_bandwidth(10),...

% J2(10),’pb’,three_db_bandwidth(11),J2(11),’xb’,...

% three_db_bandwidth(12),J2(12),’+b’,’MarkerSize’,10)

% xlabel(’Yaw Rate Bandwidth, Hz’,’fontweight’,’bold’)

% ylabel(’J2 Normalized’,’fontweight’,’bold’)

% grid

subplot(2,2,2) plot(Ay_phase_1Hz(1),J2(1),’^k’,Ay_phase_1Hz(2),J2(2),’vk’,...

Ay_phase_1Hz(3),J2(3),’k’,...

Ay_phase_1Hz(5),J2(5),’hb’,Ay_phase_1Hz(10),J2(10),’pb’,...

147 Ay_phase_1Hz(11),J2(11),’xb’,Ay_phase_1Hz(12),J2(12),’+b’,...

Ay_phase_1Hz(6),J2(6),’or’,Ay_phase_1Hz(7),J2(7),’sr’,...

Ay_phase_1Hz(8),J2(8),’.r’,Ay_phase_1Hz(9),J2(9),’*r’,...

’MarkerSize’,10) legend(’60/40, OE Tire’,’60/40, Tire 1’,’60/40, Tire 2’,...

’60/40, Tire 3’,’55/45, OE Tire’,’55/45, Tire 1’,...

’55/45, Tire 2’,’55/45, Tire 3’,’50/50, OE Tire’,...

’50/50, Tire 1’,’50/50, Tire 2’,’50/50, Tire 3’,...

’location’,’best’) xlabel(’Lateral Acceleration Phase Lag at 1 Hz, deg’,...

’fontweight’,’bold’) ylabel(’J2 Normalized’,’fontweight’,’bold’) grid

% subplot(3,2,2)

% plot(three_db_bandwidth(1),J1(1),’^k’,three_db_bandwidth(2),...

% J1(2),’vk’,three_db_bandwidth(3),J1(3),’

% three_db_bandwidth(4),J1(4),’>k’,three_db_bandwidth(5),J1(5),...

% ’hb’,three_db_bandwidth(6),J1(6),’or’,three_db_bandwidth(7),...

% J1(7),’sr’,three_db_bandwidth(8),J1(8),’.r’,...

% three_db_bandwidth(9),J1(9),’*r’,three_db_bandwidth(10),...

% J1(10),’pb’,three_db_bandwidth(11),J1(11),’xb’,...

% three_db_bandwidth(12),J1(12),’+b’,’MarkerSize’,10)

% xlabel(’Yaw Rate Bandwidth, Hz’,’fontweight’,’bold’)

% ylabel(’J1 Normalized’,’fontweight’,’bold’)

% grid

148 % subplot(2,2,1)

% plot(SS_yaw_rate_gain(1),J1(1),’^k’,SS_yaw_rate_gain(2),J1(2),...

% ’vk’,SS_yaw_rate_gain(3),J1(3),’

% ’>k’,SS_yaw_rate_gain(5),J1(5),’hb’,SS_yaw_rate_gain(6),J1(6),...

% ’or’,SS_yaw_rate_gain(7),J1(7),’sr’,SS_yaw_rate_gain(8),J1(8),...

% ’.r’,SS_yaw_rate_gain(9),J1(9),’*r’,SS_yaw_rate_gain(10),J1(10),...

% ’pb’,SS_yaw_rate_gain(11),J1(11),’xb’,SS_yaw_rate_gain(12),...

% J1(12),’+b’,’MarkerSize’,10)

% xlabel(’Steady State Yaw Rate Gain, 1/s’,’fontweight’,’bold’)

% ylabel(’J1 Normalized to OE Tire’,’fontweight’,’bold’)

% grid

subplot(2,2,1) plot(three_db_bandwidth(1),J2(1),’^k’,three_db_bandwidth(2),...

J2(2),’vk’,three_db_bandwidth(3),J2(3),’

three_db_bandwidth(4),J2(4),’>k’,three_db_bandwidth(5),J2(5),...

’hb’,three_db_bandwidth(6),J2(6),’or’,three_db_bandwidth(7),...

J2(7),’sr’,three_db_bandwidth(8),J2(8),’.r’,...

three_db_bandwidth(9),J2(9),’*r’,three_db_bandwidth(10),...

J2(10),’pb’,three_db_bandwidth(11),J2(11),’xb’,...

three_db_bandwidth(12),J2(12),’+b’,’MarkerSize’,10) xlabel(’Yaw Rate Bandwidth, Hz’,’fontweight’,’bold’) ylabel(’J2 Normalized to OE Tire’,’fontweight’,’bold’) grid

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure

149 stem3(SS_yaw_rate_gain(1),three_db_bandwidth(1),J1(1),’^k’) hold on stem3(SS_yaw_rate_gain(2),three_db_bandwidth(2),J1(2),’vk’) hold on stem3(SS_yaw_rate_gain(3),three_db_bandwidth(3),J1(3),’k’) hold on stem3(SS_yaw_rate_gain(5),three_db_bandwidth(5),J1(5),’hb’) hold on stem3(SS_yaw_rate_gain(10),three_db_bandwidth(10),J1(10),’pb’) hold on stem3(SS_yaw_rate_gain(11),three_db_bandwidth(11),J1(11),’xb’) hold on stem3(SS_yaw_rate_gain(12),three_db_bandwidth(12),J1(12),’+b’) hold on stem3(SS_yaw_rate_gain(6),three_db_bandwidth(6),J1(6),’or’) hold on stem3(SS_yaw_rate_gain(7),three_db_bandwidth(7),J1(7),’sr’) hold on stem3(SS_yaw_rate_gain(8),three_db_bandwidth(8),J1(8),’.r’) hold on stem3(SS_yaw_rate_gain(9),three_db_bandwidth(9),J1(9),’*r’) hold on grid on xlabel(’Steady State Yaw Rate Gain, 1/s’,’fontweight’,’bold’) ylabel(’Yaw Rate Bandwidth, Hz’,’fontweight’,’bold’) zlabel(’J1 Normalized’,’fontweight’,’bold’)

150 hold off

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure stem3(Ay_phase_1Hz(1),three_db_bandwidth(1),J2(1),’^k’) hold on stem3(Ay_phase_1Hz(2),three_db_bandwidth(2),J2(2),’vk’) hold on stem3(Ay_phase_1Hz(3),three_db_bandwidth(3),J2(3),’k’) hold on stem3(Ay_phase_1Hz(5),three_db_bandwidth(5),J2(5),’hb’) hold on stem3(Ay_phase_1Hz(10),three_db_bandwidth(10),J2(10),’pb’) hold on stem3(Ay_phase_1Hz(11),three_db_bandwidth(11),J2(11),’xb’) hold on stem3(Ay_phase_1Hz(12),three_db_bandwidth(12),J2(12),’+b’) hold on stem3(Ay_phase_1Hz(6),three_db_bandwidth(6),J2(6),’or’) hold on stem3(Ay_phase_1Hz(7),three_db_bandwidth(7),J2(7),’sr’) hold on stem3(Ay_phase_1Hz(8),three_db_bandwidth(8),J2(8),’.r’) hold on stem3(Ay_phase_1Hz(9),three_db_bandwidth(9),J2(9),’*r’) hold on grid on

151 xlabel(’Lateral Accel. Phase Lag, deg’,’fontweight’,’bold’) ylabel(’Yaw Rate Bandwidth, Hz’,’fontweight’,’bold’) zlabel(’J2 Normalized’,’fontweight’,’bold’) hold off

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure stem3(Ay_phase_1Hz(1),three_db_bandwidth(1),J3(1),’^k’)

hold on stem3(Ay_phase_1Hz(2),three_db_bandwidth(2),J3(2),’vk’)

hold on stem3(Ay_phase_1Hz(3),three_db_bandwidth(3),J3(3),’k’) hold on stem3(Ay_phase_1Hz(5),three_db_bandwidth(5),J3(5),’hb’) hold on stem3(Ay_phase_1Hz(10),three_db_bandwidth(10),J3(10),’pb’) hold on stem3(Ay_phase_1Hz(11),three_db_bandwidth(11),J3(11),’xb’) hold on stem3(Ay_phase_1Hz(12),three_db_bandwidth(12),J3(12),’+b’) hold on stem3(Ay_phase_1Hz(6),three_db_bandwidth(6),J3(6),’or’) hold on stem3(Ay_phase_1Hz(7),three_db_bandwidth(7),J3(7),’sr’) hold on stem3(Ay_phase_1Hz(8),three_db_bandwidth(8),J3(8),’.r’)

152 hold on stem3(Ay_phase_1Hz(9),three_db_bandwidth(9),J3(9),’*r’) hold on grid on xlabel(’Lateral Accel. Phase Lag, deg’,’fontweight’,’bold’) ylabel(’Yaw Rate Bandwidth, Hz’,’fontweight’,’bold’) zlabel(’J3 Normalized’,’fontweight’,’bold’) hold off

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% spider plot for all cases

legend1={’60/40, OE Tire’,’60/40, Tire 1’,’60/40, Tire 2’,...

’60/40, Tire 3’,’55/45, OE Tire’,’50/50, OE Tire’,...

’50/50, Tire 1’,’50/50, Tire 2’,’50/50, Tire 3’,...

’55/45, Tire 1’,’55/45, Tire 2’,’55/45, Tire 3’,}; axes_label = {’Lateral Accel Phase @ 1 Hz’ ’(deg)’;...

’Yaw Rate Bandwidth’ ’(Hz)’; ’SS Yaw Rate Gain’ ’(deg/s)/deg’; ...

’Yaw Rate Damping Ratio’ “};

[f, o] = spider(data12,“,[-70 -30 ;0 3.5;0.0 0.6;0 0.6],...

axes_label,legend1);

%% plot the spider plot for IVJD

data=[Ay_phase_1Hz(1) three_db_bandwidth(1) SS_yaw_rate_gain(1)...

153 damping_coeff(1); Ay_phase_1Hz(5) three_db_bandwidth(5)...

SS_yaw_rate_gain(5) damping_coeff(5); Ay_phase_1Hz(6) ...

three_db_bandwidth(6) SS_yaw_rate_gain(6) damping_coeff(6)]’;

legend1={’60/40 Weight Dis. (OE Tire)’,...

’55/45 Weight Dis. (OE Tire)’,...

’50/50 Weight Dis. (OE Tire)’};

axes_label = {’Lateral Accel Phase @ 1 Hz’ ’(deg)’;...

’Yaw Rate Bandwidth’ ’(Hz)’; ’SS Yaw Rate Gain’ ’(deg/s)/deg’;...

’Yaw Rate Damping Ratio’ “};

[f, o] = spider(data1,“,[-70 -30 ;0 2.5;0.0 0.6;0 0.6],...

axes_label,legend1);

%% spider plot for IJVD paper

legend1={’OE Tire’,’Tire 1’,’Tire 2’,’Tire 3’}; axes_label = {’Lateral Accel Phase @ 1 Hz’ ’(deg)’;...

’Yaw Rate Bandwidth’ ’(Hz)’; ’SS Yaw Rate Gain’ ’(deg/s)/deg’;...

’Yaw Rate Damping Ratio’ “};

[f, o] = spider(data,“,[-70 -30 ;0 3.5;0.0 0.6;0 0.6],...

axes_label,legend1); print -djpeg -r500 spider_IJVD

154 F.6 Lane_Change.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%

% CarSimLaneChange4.m

% Redefined del(Y) = Ycar - Ycourse

% Also d(X)/dt = vx*cos(beta + psi)

%%%%%%%%%%%%%%%%%%%%%%%%%%%

close all

clear

clc

format short g

format compact

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Lane Change Driver Parameters

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Kst=13.8; % Hand wheel/Road wheel ratio

K= 0.75; %[0.55 0.6 0.65 0.7 0.75 0.8 0.85];

lead=0.2; % [0.1 0.2 0.3 0.4 0.5]; %

tau=0.1; % time delay, sec

TD=0.1; % time lag, sec

vkph1=78; % Speed, kph MUST ALSO BE IN CARSIM !!

% q1=0.5; % weighting for J1 for selection of driver model

% q2=0.25; % weighting for J2 for selection of driver model

% q3=0.25; % weighting for J3 for selection of driver model

155 q1=0.75; % weighting for J1 for selection of driver model q2=0.125; % weighting for J2 for selection of driver model q3=0.125; % weighting for J3 for selection of driver model

vmps1=vkph1/3.6; lookaheadtime=1.5; % preview time for the driver model elm=lookaheadtime*vmps1; % preview time converted into distance

LookAhead=elm;

fTrack=1.453; % front track width of the Mini rTrack=1.475; % rear track width of the Mini aTrack=(fTrack+rTrack)/2; % average of track width of the Mini f/r hTrack=aTrack/2; % half track width of the Mini disp(’ ’)

% loop for determining the best parameters for gain and lead

for i=1:length(K)

for j=1:length(lead)

Kd=K(i);

TL=lead(j);

xm1=0:15;

Y1=zeros(size(xm1)); % Y, m

Psi01=zeros(size(xm1)); % rad

156 xm2=16:45;

Y2=(3.5/2)*(1 - cos((pi/30)*(xm2-15))); % Y, m

Psi02=atan((3.5*pi/60)*sin((pi/30)*(xm2-15))); % rad

xm3=46:70;

Y3=3.5*ones(size(xm3)); % Y, m

Psi03=zeros(size(xm3)); % rad

xm4=71:95;

Y4=(3.5/2)*(1 + cos((pi/25)*(xm4-70))); % Y, m

Psi04=-atan((3.5*pi/50)*sin((pi/25)*(xm4-70))); % rad

xm5=96:200;

Y5=zeros(size(xm5)); % Y, m

Psi05=zeros(size(xm5)); % rad

Ycourse=[Y1 Y2 Y3 Y4 Y5]; % Y of course, m

Xcourse=[xm1 xm2 xm3 xm4 xm5]; % X, m

Psi0=[Psi01 Psi02 Psi03 Psi04 Psi05]; % rad

% Define course boundaries, ISO TR 3888

% lane widths

% section 1 (0-15 m): width = 6.1 ft (1.86 m)

% section 2 (15-45 m): width = undefined

% section 3 (45-70 m): width = 6.6 ft (2.01 m)

% section 4 (70-95 m): width = undefined

% section 5&6 (95-125 m): width = 7.1 ft (2.15 m)

LC1x=[0 15]; % meters

157 LC1u=[(1.1*aTrack+0.25)/2 (1.1*aTrack+0.25)/2];

% meters (formula for lane width in section 1 is:...

% meters 1.1*B + 0.25 where B is width of the car)

LC1L=[-(1.1*aTrack+0.25)/2 -(1.1*aTrack+0.25)/2];

LC3x=[45 70]; % meters

LC3u=[3.5+(1.2*aTrack+0.25)/2 3.5+(1.2*aTrack+0.25)/2];

% meters (formula for lane width in section 3 is:...

% meters 1.2*B + 0.25 where B is the width of the car)

LC3L=[3.5-(1.2*aTrack+0.25)/2 3.5-(1.2*aTrack+0.25)/2];

LC5x=[95 125]; % meters

LC5u=[(1.3*aTrack+0.25)/2 (1.3*aTrack+0.25)/2];

% meters (formula for lane width in section 5 is:...

% meters 1.3*B + 0.25 where B is the width of the car)

LC5L=[-(1.3*aTrack+0.25)/2 -(1.3*aTrack+0.25)/2];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t0=0; % Start time, sec tf=10; % End time, sec

decimate=10001;

sim(’Genta_Lead_LaneChange’)

% Redefined Del(Y) = Ycar - Ycourse

% clc

% disp(’ISO Lane Change with Mini’) fprintf(’v, Initial Speed, kph %.2f \n’,vmps_nom(1)*3.6)

158 fprintf(’K, Gain, SWA/Psi, deg/deg %.2f \n’,Kd) fprintf(’tau, Time lag, sec %.3f \n’,tau) fprintf(’elm, Prediction distance, m %.2f \n’,elm) fprintf(’Prediction Time, sec %.2f \n’,lookaheadtime) fprintf(’Lead Time, sec %.2f \n’,TL)

% My output parameters from Simulink are:

% x1

% x2

% x3

% x4

% ayg

% Ycar

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Definition of States

% x1 % beta in deg for actual Psi0

% x2 % yaw rate in deg/s for actual Psi0

% x3 % yaw or heading angle psi in deg

% x4 % del-y (lat disp), m for actual Psi0 (body axes)

% x5 % SWA, deg

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

disp(’ ’) disp(’MAXIMUM VALUES’) disp(’ ’) disp(’Max(abs)values of:’) disp(’Beta,deg Yaw Rate,deg/s Y-Ycar,m ayg, Gs’) disp([max(abs(x1)) max(abs(x2)) max(abs(Y-Ycar)) max(abs(ayg))])

159 disp(’ ’) disp(’ ’) disp(’Task Performance Values’) disp(’ ’) disp(’J1 = int(del-y), cm^2*sec % Measure of lateral deviation’) disp(’ % from path’) disp(’ ’) disp(’J2 = int(SWA^2)*dt, deg^2,sec % Measure of physical workload’) disp(’ ’) disp(’J3 = int[d(SWA)/dt]^2*dt, (deg/s)^2*sec’) disp(’ % Measure of mental workload’) disp(’ % involves use of lead by driver’) disp(’ ’) disp(’ ’) fprintf(’J1, m^2*sec %.5e \n’,J1(end)) fprintf(’J2, rad^2*sec %.5e \n’,J2(end)) fprintf(’J3, (rad/s)^2*sec %.5e \n’,J3(end)) disp(’ ’)

vkph=vmps_nom(end)*3.6;

J11(j,i)=J1(end);

J22(j,i)=J2(end);

J33(j,i)=J3(end);

end end

Ji=[J11 J22 J33];

160 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% determine whether the vehicle successfully completes

% the lane change must be commented

% appropriately depending on speed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

D=zeros(1,length(WC_L1_Y)); for i=1:length(WC_L1_Y)

if WC_L1_Y(i)>=4.5034 %

D(i)=1;

elseif WC_R1_Y(i)<=-1.0766 %

D(i)=1;

elseif WC_R1_Y(i)>=-1.0766 %

D(i)=0;

elseif WC_L1_Y(i)<=4.5034 % f

D(i)=0;

elseif WC_R1_Y(1820)<=2.4966 % for 80 kph

D(i)=1; %

elseif WC_L1_Y(4316)>=1.0766 % for 80 kph

D(i)=1;

end end

sum=sum(D); disp(’Did Vehicle Successfully Complete Double Lane Change?’)

161 disp(’If value is = to 0 then successful if’) disp(’value is > than 0 then fail’) disp(sum)

% Ypsiup=(Ycar+hTrack).*(sin(x3/57.3)+cos(x3/57.3));

% Ypsidown=(Ycar-hTrack).*(sin(x3/57.3)+cos(x3/57.3));

% normalized J1, J2, J3 to the minimum for that set

for i=1:length(K)

for j=1:length(lead)

J1norm(j,i)=J11(j,i)/min(min(J11));

J2norm(j,i)=J22(j,i)/min(min(J22));

J3norm(j,i)=J33(j,i)/min(min(J33));

end end

Jtotal=J1norm*q1+J2norm*q2+J3norm*q3; figure surf(K(1:6),lead(1:4),Jtotal(1:4,1:6)) xlabel(’Driver Gain’,’fontweight’,’bold’) ylabel(’Driver Lead’,’fontweight’,’bold’) zlabel(’Jtotal normalized’,’fontweight’,’bold’) print -djpeg -r300 Jtotal_Driver_Selection

figure plot(X,x4.*cos(x3)+Y,’linewidth’,1.5)

162 plot(WC_L1_X,WC_L1_Y,’k--’,WC_R1_X,WC_R1_Y,’k--’,’linewidth’,1.5) legend(’Tire Path’) %’Desired CG Course’, hold on plot(LC1x,LC1u,LC1x,LC1L,...

LC3x,LC3u,LC3x,LC3L,...

LC5x,LC5u,LC5x,LC5L,’linewidth’,3) hold off grid ylabel(’Y, m’,’fontweight’,’bold’) xlabel(’X, m’,’fontweight’,’bold’) title(’ISO TR 3888 Lane Change Course’,’fontweight’,’bold’) print -djpeg -r300 DLC_Tire_Path

figure subplot(221),plot(t,Y-Ycar,’linewidth’,1.5) grid xlabel(’t, sec’,’fontweight’,’bold’) ylabel(’Y-Ycar, m’,’fontweight’,’bold’)

subplot(222),plot(t,x1,’linewidth’,1.5) grid xlabel(’Time, sec’,’fontweight’,’bold’) ylabel(’Beta, deg’,’fontweight’,’bold’)

subplot(223),plot(t,SWA,’linewidth’,1.5) grid xlabel(’Time, sec’,’fontweight’,’bold’)

163 ylabel(’SWA, deg’,’fontweight’,’bold’)

subplot(224),plot(t,x2,’linewidth’,1.5) grid xlabel(’Time, sec’,’fontweight’,’bold’) ylabel(’r, deg/s’,’fontweight’,’bold’) title(’Yaw Rate’,’fontweight’,’bold’) print -djpeg -r300 DLC_Response_Vals

figure subplot(211),plot(t,ayg,’linewidth’,1.5) xlabel(’Time, sec’,’fontweight’,’bold’) ylabel(’Lat Acc, Gs’,’fontweight’,’bold’) grid

subplot(212),plot(X,ayg,’linewidth’,1.5) xlabel(’X Distance, m’,’fontweight’,’bold’) ylabel(’Lat Acc, Gs’,’fontweight’,’bold’) grid print -djpeg -r300 DLC_Lat_accel

figure subplot(211),plot(t,vmps_nom*3.6,’linewidth’,1.5) xlabel(’Time, sec’,’fontweight’,’bold’) ylabel(’Speed, kph’,’fontweight’,’bold’) grid

subplot(212),plot(X,vmps_nom*3.6,’linewidth’,1.5)

164 xlabel(’X Distance, m’,’fontweight’,’bold’) ylabel(’Speed, kph’,’fontweight’,’bold’) grid print -djpeg -r300 DLC_Speed

figure subplot(211),plot(t,x4.*cos(x3)+Y, t,Y,’--’,’linewidth’,1.5) subplot(211),plot(t,Ycar, t,Y,’--’,’linewidth’,1.5) xlabel(’Time, sec’,’fontweight’,’bold’) ylabel(’y, m’,’fontweight’,’bold’) grid legend(’Ycar’,’Ycourse’)

subplot(212),plot(t,Psi*57.3, t,x3,’--’,’linewidth’,1.5) xlabel(’Time, sec’,’fontweight’,’bold’) ylabel(’Heading Angle, deg’,’fontweight’,’bold’) grid legend(’Psi0-com’,’Psi’) print -djpeg -r300 DLC_Heading_Angle

figure plot(X,Y,’linewidth’,1.5) %X,Ycar,’--’, legend(’Desired CG Path’,’Actual CG Path’) hold on plot(LC1x,LC1u,LC1x,LC1L,...

LC3x,LC3u,LC3x,LC3L,...

LC5x,LC5u,LC5x,LC5L,’linewidth’,3)

165 hold off grid ylabel(’Y, m’,’fontweight’,’bold’) xlabel(’X, m’,’fontweight’,’bold’) title(’ISO TR 3888 Lane Change Course’,’fontweight’,’bold’)

% print -djpeg -r300 DLC_CG_Path

166 References

[1] Anon., “Test Track for a Severe Lane Change Maneuver,” 1-ed, International Standard Organization, ISO 3888, 1975.

[2] Mimuro, T., Ohsaki, M., Yasunaga, H., and Satoh, K., “Four Parameter Evaluation Method of Lateral Transient Response,” SAE Paper 901734, Society of Automotive Engineers, Warrendale, PA, September 1990.

[3] Pacejka, H., Tire and Vehicle Dynamics, 1-ed, Society of Automotive Engineers, War- rendale, PA, 2002.

[4] McKibben, D., and Frey,N., “2007 BMW Mini Examination of Suspension & Steering Kinematics & Compliance”, Michelin Report FW576, March 2007.

[5] Rhyne, T., “Flat Track Test Pacejka Data for BMW MINI OEM Tires”, Test Ranking No. J7H46030, Michelin America Research Center, April, 2007

[6] Powell, R., Khekare, A., Law, E. H., Ayalew, B., “Transient Handling Analysis of a 2007 BMW Mini Equipped with TWEELs: Comparison of Tests & Simulation and Pa- rameter Studies,” Clemson University, Department of Mechanical Engineering Report TR-08-119-ME-MMS, March 2009.

[7] Crolla, D., Chen, D., Whitehead, J., and Alstead, C., “Vehicle Handling Assessment Using a Combined Subjective-Objective Approach,” SAE Paper 980226, Society of Automotive Engineers, Warrendale, PA, February 1998.

[8] King, R., Crolla, D., Ash, H., and Whitehead, J., “Identiﬁcation of Subjective- Objective Vehicle Handling Links Using Neural Networks for the Foresight Vehicle”, SAE Paper 2002-01-1126, Society of Automotive Engineers, Warrendale, PA, March 2002.

[9] Lincke, W., Richter, B., and Schmidt, R., “Simulation and measurement of driver vehicle handling performance.” SAE Paper 730489, Society of Automotive Engineers, Warrendale, PA, May 1973.

[10] Meetings with Dr. Rhyne, Steve Cron, BMW representatives, and Clemson representatives, June 2008-September 2008

167 [11] Bergman, W., “Correlation Between Open and Closed Loop Vehicle Handling Mea- surements,” SAE Paper 800846, Society of Automotive Engineers, Warrendale, PA, June 1980.

[12] Good, M., “Sensitivity of Driver-Vehicle Performance to Vehicle Characteristics Re- vealed in Open-loop Tests,” Vehicle System Dynamics, Volume 6, Issue 4, pp. 245-277, Netherlands, October 1977.

[13] McRuer, D. and Klein, R., Automobile Controllability-Driver/Vehicle Response for Steering Control, Vol. I Summary Report, U.S. Dept. of Transportation, DOT HS-801- 407, February 1975.

[14] Lincke, W., Richter, B., and Schmidt, R., “Simulation and Measurement of Driver Vehicle Handling Performance,” SAE Paper 730489, Society of Automotive Engineers, Warrendale, PA, May 1973.

[15] Genta, G., Motor Vehicle Dynamics: Modeling and Simulation, 1-ed, World Scientiﬁc, Singapore, 1997.

[16] Horiuchi, S. and Yuhara, N., “An Analytical Approach to the Prediction of Handling Qualities of Vehicles With Advanced Steering Control System Using Multi-Input Driver Model,” Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control, Volume 122, Issue 3, pp. 490-497, United States, September 2000.

[17] Oscarsson, M. “Variable Vehicle Dynamics Design-Objective Design Methods,” Mas- ter’s Thesis performed in Vehicular Systems, Department of Electrical, Linkopings University, July 2003.

[18] Law, E. H., ME 453/653 Class Notes, Clemson University, Fall 2007.

[19] Anon., “CarSim 8 Math Models,” Mechanical Simulation Corporation. http://www.carsim.com/downloads/pdf/Math_Models_C80.pdf. June 2009.

[20] Henderson, J., “2007 BMW Mini, TWEEL Auto Study, Project ID A7255A, Dry Ob- jective Handling Test Report”, Test No. 172899, Michelin America Research Center, April2007.

[21] Sutton, R., Modelling Human Operators in Control System Design, Research Studies Press LTD. Taunton, Somerset, England. 1990.

[22] Law, E. H., “Development of Input Parameters for a Simulation of the Vertical Ride Response of a Model Year 2007, BMW Mini REVISED,” Clemson Report, Report No. TR-07-120-ME-MMS, December 28, 2007.

[23] Law, E. H., “Development of Input Parameters for a Simulation of the Lateral Dynamic Response of a Model year 2000, C-5 Corvette,” Clemson Report, Report No. TR-00- 102-ME-MMS, August 2000.

[24] Rhyne, T. XYZ Rigidity Test for BMW Mini OEM Tires, Test No. 172421, Michelin America Research Center, April 2007.

168 [25] Day, Terry D., Metz, L. Daniel, “The Simulation of Driver Inputs Using a Vehicle Driver Model,” SAE Paper 2000-01-1313, Society of Automotive Engineers, Warrendale, PA, March 2000.

169